Short answer
No. Muslims do believe in the Mahdi. Sound hadith describe a man from the Prophet's (ﷺ) family. Allah (﷾) will raise him up near the end of time to fill the earth with justice. But the Qur'an and the sound Sunnah do not describe an imam who has lived hidden for over a thousand years. The Qur'an says Allah (﷾) has completed the religion, and its guidance is in a Book every believer can read. It calls Muhammad (ﷺ) the Seal of the Prophets. Ali himself said his family had no book besides the Qur'an, apart from the understanding Allah (﷾) gives a Muslim and one sheet of rulings. We love the Prophet's (ﷺ) family, and we wait for the Mahdi as the hadith describe him.
Full explanation
The Mahdi in the sound hadith
Sunan Abi Dawud records sound reports on the Mahdi. Here are two of them. In the first, Ali himself reports the Prophet's (ﷺ) words. If only one day of the world remained, Allah (﷾) would raise up a man from his family. He would fill the earth with justice as it was filled with oppression. In the second, Umm Salama says the Prophet (ﷺ) said the Mahdi will be of his family, of the descendants of Fatima. Muslims believe this. A further report in Abu Dawud, graded hasan sahih, describes the same man as one who bears the Prophet's (ﷺ) name; one wording adds that his father's name is the Prophet's father's name. The reports speak of a man whom Allah (﷾) will raise up. We say what they say, and no more.
Ali ibn Abi Talib reports that the Prophet (ﷺ) said: if only one day of this world remained, Allah would raise up a man from my family who would fill the earth with justice as it has been filled with oppression.
Umm Salama reports that the Prophet (ﷺ) said: the Mahdi will be of my family, of the descendants of Fatima.
Abdullah ibn Mas'ud reports that the Prophet (ﷺ) said: if only one day of this world remained, Allah would lengthen that day until He raised up in it a man who belongs to me or to my family, whose father's name is the same as my father's, who will fill the earth with equity and justice as it has been filled with oppression and tyranny. In another wording the world will not pass away before the Arabs are ruled by a man of my family whose name is the same as mine.
The religion has been completed
The Qur'an says that Allah (﷾) perfected the religion and completed His favour. Its guidance does not wait on a guide whom no one has been able to reach for over a thousand years. Prophethood ended with Muhammad (ﷺ), the Seal of the Prophets. After the messengers, people have no plea against Allah (﷾), because the Message has reached them. And Allah (﷾) promised to guard the Message Himself. The guidance is in a Book that every believer can read.
“This day have I perfected your religion for you, completed My favour upon you, and have chosen for you Islam as your religion.”
“Muhammad is not the father of any of your men, but (he is) the Messenger of Allah, and the Seal of the Prophets: and Allah has full knowledge of all things.”
“Messengers who gave good news as well as warning, that mankind, after (the coming) of the messengers, should have no plea against Allah: For Allah is Exalted in Power, Wise.”
“We have, without doubt, sent down the Message; and We will assuredly guard it (from corruption).”
The unseen belongs to Allah (﷾)
The Qur'an says none in the heavens or on the earth knows what is hidden except Allah (﷾). He shows His secrets only to a messenger He chooses. The Prophet (ﷺ) was told to say that he did not know the unseen. Ali was asked whether the family had any book besides the Qur'an. He said no, except the understanding Allah (﷾) gives a Muslim and one sheet of rulings on blood-money, captives and retaliation.
“Say: None in the heavens or on earth, except Allah, knows what is hidden: nor can they perceive when they shall be raised up (for Judgment).”
“"He (alone) knows the Unseen, nor does He make any one acquainted with His Mysteries,- "Except a messenger whom He has chosen: and then He makes a band of watchers march before him and behind him,”
“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."”
Abu Juhayfa asked Ali whether the family had any book besides the Qur'an. Ali said no: only the Book of Allah, the understanding Allah gives a Muslim, and what is on this sheet. The sheet held rules on blood-money, on ransoming captives, and a ruling on retaliation (qisas).
Long life: what Allah (﷾) has told us, and what He has not
Allah (﷾) can do all things, so the question is what He told us. Muslims believe that Jesus was raised up by Allah (﷾), and will return. The Qur'an says Allah (﷾) raised him, and the Prophet (ﷺ) swore that the son of Mary will descend. Belief in the unseen must rest on revelation. Allah (﷾) told us of Jesus; He has told us nothing of an imam kept alive and hidden for centuries.
“Nay, Allah raised him up unto Himself; and Allah is Exalted in Power, Wise;-”
Abu Huraira reports that the Prophet (ﷺ) swore that the son of Mary will soon descend among the Muslims as a just ruler. He will break the cross, kill the swine and end the jizya, and wealth will be so plentiful that no one will accept it. Abu Huraira then recited Qur'an 4:159: none of the People of the Book but will believe in him before his death.
Where the Prophet (ﷺ) pointed us
At Khumm the Prophet (ﷺ) said he was leaving two weighty things. The first is the Book of Allah (﷾), with right guidance and light, and he told the people to hold fast to it. The second is the people of his household, and he reminded the people of their duty to them. Zayd ibn Arqam, who reported it, named them as the families of Ali, Aqil, Ja'far and al-Abbas. That duty is love, honour and care.
Zayd ibn Arqam reports that at a watering place called Khumm, between Mecca and Medina, the Prophet (ﷺ) praised Allah and said he was leaving two weighty things among the people: the Book of Allah, which holds right guidance and light, so hold fast to it; and the people of his household, reminding the people of their duties to the members of his family. Zayd said the Prophet's (ﷺ) wives are of his household, but the household meant here are those to whom charity is forbidden: the family of Ali, of Aqil, of Ja'far and of al-Abbas.
Perspectives
Some Muslims believe the last imam is alive and hidden
Some Muslims believe that Allah never leaves the earth without a guide He has appointed. They hold that the last of these guides was born long ago and is hidden by Allah's will until Allah allows his return. They bring these arguments.
The earth is never without a proof from Allah, so a guide must exist in every age.
Rebuttal
The Qur'an names the proof: the messengers, after whom people have no plea against Allah (﷾). It also says Allah (﷾) Himself guards the Message, so the proof remains in the Book.
The Prophet (ﷺ) spoke of twelve caliphs, and the twelfth is the hidden imam.
Rebuttal
The report says there will be twelve caliphs, all from Quraysh. It names none of them and says nothing about anyone being hidden. A caliph, in this report, is a leader of the community. Other wordings say Islam will stay mighty under these twelve. They describe leaders who rule, not one who is hidden.
Allah can keep a man alive as long as He wills, as He did with Jesus.
Rebuttal
Allah (﷾) can do all things. Our belief about Jesus rests on the Qur'an and on the Prophet's (ﷺ) words about his return. About a hidden imam, neither speaks.
The hadith of the Mahdi from the Prophet's (ﷺ) family describe the hidden imam.
Rebuttal
Muslims believe in the Mahdi of these reports. They say Allah (﷾) will raise him up near the end of time. One report gives his name as the Prophet's (ﷺ) name and, in one wording, his father's name as the Prophet's father's name. None says that he was born long ago or is hidden.
Without a present, divinely guided imam, believers have no one to settle their disputes.
Rebuttal
The Qur'an says: if you differ in anything, refer it to Allah (﷾) and His Messenger. Ali himself said his family had no book besides the Qur'an. They had only the understanding Allah (﷾) gives a Muslim, and one sheet of rulings.
The Islamic perspective
The Mahdi of the sound hadith is a man of the Prophet's (ﷺ) family, raised up near the end of time. The religion is complete, and Muhammad (ﷺ) is the last prophet. Disputes go back to Allah (﷾) and His Messenger.
Allah (﷾) has perfected the religion and completed His favour.
Muhammad (ﷺ) is the Seal of the Prophets.
The Mahdi is a man of the Prophet's (ﷺ) family. Allah (﷾) will raise him up near the end of time.
Ali said his family had no book besides the Qur'an. They had only the understanding Allah (﷾) gives a Muslim, and one sheet of rulings.
Knowledge of the unseen belongs to Allah (﷾) alone.
In every dispute, believers return to Allah (﷾) and His Messenger.