Short answer
No. Calling on Allah (﷾) in need is worship, and worship belongs to Allah (﷾) alone. In the Qur'an Allah says: "Call on Me; I will answer your (Prayer)". The Prophet (ﷺ) taught that supplication is worship. He called on his own Lord, and Allah (﷾) told him to say that he could not bring anyone harm or right conduct. Ali narrated how the Prophet (ﷺ) prayed, and the prayers he passed on in the sound reports call on Allah (﷾) alone. Ali and the Prophet's (ﷺ) family loved Allah (﷾) and His Messenger (ﷺ), and we love them for it. Asking a living person to pray for you, or to help in what he can do, is fine. Calling on the dead or the absent for what only Allah (﷾) can give is not. Intercession is real, and it is in Allah's (﷾) gift alone.
Full explanation
Calling on Allah (﷾) is worship
Allah (﷾) invites every servant to call on Him directly. He says He is close and that He answers. The Prophet (ﷺ) explained what this means. Supplication, or dua, is itself worship. He recited the verse to show it. Worship is for Allah (﷾) alone, so dua is for Allah (﷾) alone.
“And your Lord says: "Call on Me; I will answer your (Prayer): but those who are too arrogant to serve Me will surely find themselves in Hell - in humiliation!"”
“When My servants ask thee concerning Me, I am indeed close (to them): I listen to the prayer of every suppliant when he calleth on Me: Let them also, with a will, Listen to My call, and believe in Me: That they may walk in the right way.”
An-Nu'man ibn Bashir reports that the Prophet (ﷺ) said: supplication is worship. He then recited the verse in which Allah says: call on Me, I will answer you.
Every prayer says it
In every unit of prayer a Muslim says: "Thee do we worship, and Thine aid we seek". Worship and asking for aid are joined, and both go to Allah (﷾). In Surat al-Jinn Allah (﷾) says that the places of worship are for Allah (﷾), so invoke no one along with Him. In 10:106 Allah (﷾) says it even to the Prophet (ﷺ), so the rule has no exception for anyone.
“Thee do we worship, and Thine aid we seek.”
“"And the places of worship are for Allah (alone): So invoke not any one along with Allah;”
“"'Nor call on any, other than Allah;- Such will neither profit thee nor hurt thee: if thou dost, behold! thou shalt certainly be of those who do wrong.'"”
What the Prophet (ﷺ) said and did
Allah (﷾) told the Prophet (ﷺ) to say that he only calls on his Lord. He was also told to say that it is not in his power to cause harm or to bring right conduct. He was also told that the decision is not his. At Badr the Muslims were far fewer than the enemy. He did not call on any helper. He faced the qibla and raised his hands to Allah (﷾) until his cloak slipped from his shoulders.
“Say: "I do no more than invoke my Lord, and I join not with Him any (false god)." Say: "It is not in my power to cause you harm, or to bring you to right conduct."”
“Say: "I have no power over any good or harm to myself except as Allah willeth. If I had knowledge of the unseen, I should have multiplied all good, and no evil should have touched me: I am but a warner, and a bringer of glad tidings to those who have faith."”
“Not for thee, (but for Allah), is the decision: Whether He turn in mercy to them, or punish them; for they are indeed wrong-doers.”
Umar reports that at Badr the Prophet (ﷺ) looked at the enemy, about a thousand, and at his own companions, three hundred and nineteen. He faced the qibla, stretched out his hands and called on his Lord: O Allah, fulfil what You promised me. He kept on until his cloak slipped from his shoulders, and Abu Bakr put it back and said Allah would fulfil His promise. Allah then revealed that He would help with a thousand angels.
What the companions did
The Qur'an describes the Prophet's (ﷺ) companions at Badr as people who implored their Lord, and He answered them. After the Prophet (ﷺ) died, drought came. Umar asked al-Abbas to pray to Allah (﷾) for rain. Al-Abbas was the Prophet's (ﷺ) uncle, and Ali's uncle too. Umar asked a living man to pray, and he asked Allah (﷾).
“Remember ye implored the assistance of your Lord, and He answered you: "I will assist you with a thousand of the angels, ranks on ranks."”
Anas reports that after the Prophet's (ﷺ) death, whenever there was a drought, Umar ibn al-Khattab asked the Prophet's living uncle al-Abbas to pray to Allah for rain. Umar said: we used to ask our Prophet to pray to You for rain, and now we ask his uncle.
Ali called on Allah (﷾), not on himself
Ali narrated how the Prophet (ﷺ) opened his prayer. When he stood to pray, he turned his face to the One who originated the heavens and the earth, and he asked Allah (﷾) alone for forgiveness and guidance. Another supplication Ali narrated is: O Allah (﷾), guide me and keep me straight. The testimony of faith names only Allah (﷾) and His Messenger. The tashahhud (the words said while sitting in prayer) gives all praise to Allah (﷾), greets the Prophet (ﷺ) and every righteous servant of Allah (﷾), asks nothing of any of them, and ends with the same testimony. When companions greeted people by name in it, the Prophet (ﷺ) gave them one greeting that reaches every righteous servant instead. No sound report has Ali putting himself into the prayer.
Ali ibn Abi Talib reports how the Prophet (ﷺ) opened his prayer and what he said in it when he stood to pray. He began by turning his face to the One who originated the heavens and the earth, saying that his prayer, sacrifice, living and dying are for Allah, Lord of the worlds, who has no partner. He asked Allah alone for forgiveness and guidance. In bowing, rising and prostrating he addressed Allah alone, and between the tashahhud and the salutation he asked Allah to forgive his earlier and later sins.
Ali reports that the Messenger of Allah (ﷺ) told him to say: O Allah, guide me and keep me straight. He was to keep in mind the right path and the straightness of an arrow.
Umar ibn al-Khattab reports that a stranger in very white clothes, with no sign of travel on him, sat before the Prophet (ﷺ) and asked him about Islam, faith and excellence in worship. Asked about faith, the Prophet said it is to believe in Allah, His angels, His books, His messengers and the Last Day, and to believe in the divine decree, its good and its evil. Afterwards the Prophet told Umar the stranger was Gabriel, who had come to teach them their religion.
Ibn Umar reports that the Prophet (ﷺ) said Islam is built on five: to bear witness that there is no god but Allah and that Muhammad is the Messenger of Allah, to establish the prayer, to pay zakah, to perform the pilgrimage, and to fast the month of Ramadan.
Abdullah ibn Mas'ud reports that behind the Prophet (ﷺ) the companions used to say in the sitting: peace on Jibril, peace on Mika'il, peace on so-and-so. The Messenger of Allah (ﷺ) turned and said that Allah Himself is As-Salam, and taught them to say: all compliments, prayers and good things are due to Allah; peace on you, O Prophet, and Allah's mercy and blessings; peace on us and on the righteous servants of Allah. He said this greeting reaches every righteous servant in heaven and earth. Then: I testify that there is no god but Allah, and I testify that Muhammad is His servant and Messenger.
Asking the living is different
Islam does not forbid asking people for help in what they can do. Allah (﷾) tells believers to help one another in righteousness. The sons of Jacob asked their living father to seek forgiveness for them, and he said he would ask his Lord. In the Prophet's (ﷺ) lifetime people could come to him, and he would ask forgiveness for them. In each case a living person is present, and Allah (﷾) does the answering.
“O ye who believe! Violate not the sanctity of the symbols of Allah, nor of the sacred month, nor of the animals brought for sacrifice, nor the garlands that mark out such animals, nor the people resorting to the sacred house, seeking of the bounty and good pleasure of their Lord. But when ye are clear of the sacred precincts and of pilgrim garb, ye may hunt and let not the hatred of some people in (once) shutting you out of the Sacred Mosque lead you to transgression (and hostility on your part). Help ye one another in righteousness and piety, but help ye not one another in sin and rancour: fear Allah: for Allah is strict in punishment.”
“They said: "O our father! ask for us forgiveness for our sins, for we were truly at fault." He said: "Soon will I ask my Lord for forgiveness for you: for he is indeed Oft-Forgiving, Most Merciful."”
“We sent not a messenger, but to be obeyed, in accordance with the will of Allah. If they had only, when they were unjust to themselves, come unto thee and asked Allah's forgiveness, and the Messenger had asked forgiveness for them, they would have found Allah indeed Oft-returning, Most Merciful.”
Those who are called on look to Allah (﷾) themselves
Allah (﷾) says that even those people call on are themselves seeking a way of access to their Lord. This includes the righteous and the angels. They hope for His mercy and fear His wrath. They are servants like us. Allah (﷾) owns all dominion, and those invoked besides Him have not the least power. This is no comparison of the Prophet's (ﷺ) family with idols. These verses set a rule about calling on anyone besides Allah (﷾). The Qur'an adds that the dead are not made to hear by us. Allah (﷾) makes hear whom He wills.
“Say: "Call on those - besides Him - whom ye fancy: they have neither the power to remove your troubles from you nor to change them." Those whom they call upon do desire (for themselves) means of access to their Lord, - even those who are nearest: they hope for His Mercy and fear His Wrath: for the Wrath of thy Lord is something to take heed of.”
“He merges Night into Day, and he merges Day into Night, and he has subjected the sun and the moon (to his Law): each one runs its course for a term appointed. Such is Allah your Lord: to Him belongs all Dominion. And those whom ye invoke besides Him have not the least power.”
“If ye invoke them, they will not listen to your call, and if they were to listen, they cannot answer your (prayer). On the Day of Judgment they will reject your "Partnership". and none, (O man!) can tell thee (the Truth) like the One Who is acquainted with all things.”
“Nor are alike those that are living and those that are dead. Allah can make any that He wills to hear; but thou canst not make those to hear who are (buried) in graves.”
Intercession belongs to Allah (﷾)
Muslims believe the Prophet (ﷺ) will intercede on the Day of Judgement. The Qur'an says all intercession belongs to Allah (﷾), and none happens without His permission. The Prophet (ﷺ) told Abu Hurayra who would gain most from it. It is the one who said "there is no god but Allah" sincerely from the heart. The way to his intercession is therefore tawhid, the oneness of Allah (﷾), and not calling on him.
“Say: "To Allah belongs exclusively (the right to grant) intercession: to Him belongs the dominion of the heavens and the earth: In the End, it is to Him that ye shall be brought back."”
“Allah! There is no god but He,-the Living, the Self-subsisting, Eternal. No slumber can seize Him nor sleep. His are all things in the heavens and on earth. Who is there can intercede in His presence except as He permitteth? He knoweth what (appeareth to His creatures as) before or after or behind them. Nor shall they compass aught of His knowledge except as He willeth. His Throne doth extend over the heavens and the earth, and He feeleth no fatigue in guarding and preserving them for He is the Most High, the Supreme (in glory).”
“On that Day shall no intercession avail except for those for whom permission has been granted by (Allah) Most Gracious and whose word is acceptable to Him.”
Abu Huraira reports that the Prophet (ﷺ) was asked who would be the happiest of people with his intercession on the Day of Resurrection. He told Abu Huraira that, seeing his eagerness for hadith, he had expected no one to ask him about it before him. Then he said it would be the one who said "there is no god but Allah" sincerely (khalisan), from his heart.
Perspectives
Some Muslims believe the Prophet's (ﷺ) family may be asked for help
Some Muslims, out of deep love for the Prophet (ﷺ) and his family, ask them for help as a means to Allah, believing they help only by His leave. They see this as no rival to Allah. These are the main arguments they give.
Allah tells believers to seek the means (wasila) of approach to Him. The Prophet (ﷺ) and his family, they say, are that means.
Rebuttal
The verse says: do your duty to Allah (﷾), seek the means of approach, and strive. Ibn Kathir reports that the commentators explained the means as obedience and deeds that please Allah (﷾). They do not differ on this.
The Prophet (ﷺ) and the martyrs are alive with their Lord, so they hear us and can help.
Rebuttal
The verses say the martyrs are alive with their Lord, in a way we do not perceive. Allah (﷾) says those invoked do not hear the call, and if they heard, they could not answer. He makes hear whom He wills, and He tells us He is close to every caller.
The Prophet (ﷺ) spoke to the dead at the well of Badr, and said they heard him. So, they say, the dead can hear.
Rebuttal
Allah (﷾) made them hear at that moment, as He makes hear whom He wills. The Prophet (ﷺ) spoke to them; he asked them for nothing. It gives no practice of calling on the dead.
Intercession is real, so asking the intercessor directly is the shortest path.
Rebuttal
The Qur'an says intercession belongs to Allah (﷾) alone and happens only with His permission. The Prophet (ﷺ) named the sincere believer in one God (﷾) as the one who gains most from it.
People came to the Prophet (ﷺ) and he asked forgiveness for them (4:64), so this remains allowed after his death.
Rebuttal
The verse describes coming to him while he lived, when he could answer and ask for them. After his death, when drought came, Umar said they used to ask the Prophet (ﷺ) to pray for rain, and now they asked his uncle. Al-Abbas, the Prophet's (ﷺ) own uncle, prayed to Allah (﷾) for them.
The Prophet (ﷺ) taught a blind man to ask Allah by him, so asking through the Prophet (ﷺ) is allowed.
Rebuttal
In the report the man came to the Prophet (ﷺ) himself and asked him to pray for him. The Prophet (ﷺ) offered to pray, then told him how to ask. The supplication ends by asking Allah (﷾) to accept the Prophet's (ﷺ) intercession for him. It was a request to Allah (﷾), made in the Prophet's (ﷺ) presence, for his prayer. After the Prophet (ﷺ) died, the companions went to a living man, al-Abbas, and he prayed to Allah (﷾) for them.
The Qur'an shows Jesus healing, and the throne brought to Sulayman, by Allah's leave. So, they say, those near Allah can help us by His leave.
Rebuttal
These were living people, present, giving signs by Allah's (﷾) leave. The Qur'an shows no one calling on an absent or dead prophet for help.
The Islamic perspective
Dua is worship, so it belongs to Allah (﷾) alone. The Prophet (ﷺ) and his companions called on their Lord. Asking the living to pray for us is fine, and love for the Prophet's (ﷺ) family is a duty.
Supplication is worship, so it is for Allah (﷾) alone.
The Prophet (ﷺ) himself called on his Lord alone. He said it was not in his power to harm or to guide.
The companions asked a living man to pray for rain, and asked Allah (﷾).
Asking the living for help in what they can do is allowed.
Intercession belongs to Allah (﷾), and it is gained through sincere belief in His oneness.
Loving and honouring the Prophet's (ﷺ) family is a duty, and following what they followed honours them.