Short answer
It says so, but its own scriptures do not keep to it. The Dhammapada says no one can purify another. King Yama tells the wrongdoer that he alone will reap his bad deed. Yet the Buddha also says that gifts given for dead relatives reach them if they are reborn as ghosts. Another Buddhist text says the dead live on what is given here. The Buddha promises heaven to anyone who dies on pilgrimage to his shrines. He promises lasting welfare to anyone who lays garlands at his monument. So the oldest Buddhist texts teach both, and nothing in them explains how one person's gift becomes another's good. Islam has no such gap. Each soul answers for its own deeds, and Allah (﷾), who gives every reward, accepts prayer and charity for the dead.
Full explanation
Each person alone
The Dhammapada, a collection of the Buddha's sayings, says that being pure or impure is each person's own affair. No one can make another pure. It says the Buddhas only show the way: you must do the work yourselves. In the discourse called the Messengers of the Gods, King Yama tells the wrongdoer that no one else did his bad deed. He alone will feel its result.
“For it is by oneself that evil's done, one is corrupted by oneself. It's by oneself that evil's not done, one is purified by oneself. Purity and impurity are personal matters, no one can purify another.”
“You yourselves must do the work, the Realized Ones just show the way.”
“That bad deed wasn't done by your mother, father, brother, or sister. It wasn't done by friends and colleagues, by relatives and kin, by ascetics and brahmins, or by the deities. That bad deed was done by you alone, and you alone will experience the result.”
Gifts that reach the dead
A brahmin, a priest of the old Indian religion, asks the Buddha whether gifts given for the dead really help them. The Buddha answers that a gift helps a relative who has been reborn among the ghosts. Such a ghost lives on what its relatives give. If that relative is not there, other dead relatives will take the gift, and that place is never without them. A short book of verses, the Khuddakapatha, says the dead live on what is given here, as rain runs down to the plains.
“We who are known as brahmins give gifts and perform memorial rites for the dead: 'May this gift aid my departed relatives and kin. May they partake of this gift.' But does this gift really aid departed relatives and kin?”
“When their body breaks up, after death, they're reborn in the ghost realm. There they survive feeding on the food of the beings in the ghost realm. Or else they survive feeding on what friends and colleagues, relatives and kin provide them with from here. The conditions there are right, so the gift aids the one who lives there." "But worthy Gotama, who partakes of that gift if the departed relative is not reborn in that place?" "Other departed relatives reborn there will partake of that gift." "But who partakes of the gift when neither that relative nor other relatives have been reborn in that place?" "It's impossible, brahmin, it cannot happen that that place is vacant of departed relatives in all this long time.”
“The departed, the dead in that place live on what is given here. Just as water that rains on high flows down to the plains, so too what is given here aids the departed ghosts.”
Heaven for a pilgrimage, welfare for a garland
Shortly before he died, the Buddha named four places to visit: where he was born, where he awoke, where he first taught, and where he passed away. Anyone who dies on a pilgrimage to these shrines, he said, will be reborn in heaven. He asked for a monument over his remains at a crossroads. Whoever puts garlands or scent there, or bows, will have lasting welfare and happiness.
“"Ānanda, a faithful gentleman should go to see these four inspiring places. What four? Thinking: 'Here the Realized One was born!'—that is an inspiring place. Thinking: 'Here the Realized One became awakened as a supreme fully awakened Buddha!'—that is an inspiring place. Thinking: 'Here the supreme Wheel of Dhamma was rolled forth by the Realized One!'—that is an inspiring place. Thinking: 'Here the Realized One was fully quenched in the element of extinguishment with no residue!'—that is an inspiring place. These are the four inspiring places that a faithful gentleman should go to see. Faithful monks, nuns, laymen, and laywomen will come, and think: 'Here the Realized One was born!' and 'Here the Realized One became awakened as a supreme fully awakened Buddha!' and 'Here the supreme Wheel of Dhamma was rolled forth by the Realized One!' and 'Here the Realized One was fully quenched in the element of extinguishment with no residue!' Anyone who passes away while on pilgrimage to these shrines will, when their body breaks up, after death, be reborn in a good place, a heavenly realm."”
“A monument for the Realized One is to be built at the crossroads. When someone there lifts up garlands or fragrance or powder, or bows, or inspires confidence in their heart, that will be for their lasting welfare and happiness.”
Offerings to spirits
In the same discourse, the Buddha speaks of the Vajjis, a people of his time. They honour their shrines and keep up the offerings to spirits made in the past. As long as they do, he says, they can expect to grow, not to decline.
“As long as the Vajjis honor, respect, esteem, and venerate the Vajjian shrines, whether inner or outer, not neglecting the proper spirit-offerings that were given and made in the past, they can expect growth, not decline.”
Two teachings that do not fit
If no one can purify another and each person alone reaps his deed, a gift cannot pass to someone else. If it can, the fruit of a deed is not the doer's alone. In the Pali texts karma works like a law, with no one behind it. So nothing in them explains who sends a gift given here to a ghost there.
“For it is by oneself that evil's done, one is corrupted by oneself. It's by oneself that evil's not done, one is purified by oneself. Purity and impurity are personal matters, no one can purify another.”
“That bad deed wasn't done by your mother, father, brother, or sister. It wasn't done by friends and colleagues, by relatives and kin, by ascetics and brahmins, or by the deities. That bad deed was done by you alone, and you alone will experience the result.”
“When their body breaks up, after death, they're reborn in the ghost realm. There they survive feeding on the food of the beings in the ghost realm. Or else they survive feeding on what friends and colleagues, relatives and kin provide them with from here. The conditions there are right, so the gift aids the one who lives there." "But worthy Gotama, who partakes of that gift if the departed relative is not reborn in that place?" "Other departed relatives reborn there will partake of that gift." "But who partakes of the gift when neither that relative nor other relatives have been reborn in that place?" "It's impossible, brahmin, it cannot happen that that place is vacant of departed relatives in all this long time.”
One account, and a Lord who gives
The Qur'an says that no one carries another's burden, and that a person has only what he worked for. The Prophet (ﷺ) also taught that a dead person's good deeds go on through a good child who prays for him. And he taught that charity given for a dead mother reaches her. There is no clash, because the reward was never a power inside the deed itself. It is Allah's (﷾) to give, and He accepts prayer and charity for others as He wills. The Qur'an tells believers to ask forgiveness for the believing men and women.
“Namely, that no bearer of burdens can bear the burden of another; That man can have nothing but what he strives for;”
Abu Huraira reports that the Prophet (ﷺ) said that when a person dies, his deeds come to an end except three: lasting charity, knowledge that people benefit from, and a righteous child who prays for him.
Aisha reports that a man told the Prophet (ﷺ) that his mother had died suddenly. He thought she would have given charity if she had lived, and he asked whether she would receive the reward if he gave charity for her. The Prophet (ﷺ) said yes.
“Know, therefore, that there is no god but Allah, and ask forgiveness for thy fault, and for the men and women who believe: for Allah knows how ye move about and how ye dwell in your homes.”
Worship is for Allah (﷾) alone
Islam makes no offerings to spirits or to shrines. The Qur'an says the places of worship are for Allah (﷾) alone, so no one is to be invoked along with Him.
“"And the places of worship are for Allah (alone): So invoke not any one along with Allah;”
The usual answers
Qur'an 53:38-39 — Surat al-Najm (The Star), verses 38-39
“Namely, that no bearer of burdens can bear the burden of another; That man can have nothing but what he strives for;”
Cited for
Islam has the same problem. The Qur'an says man has only what he strove for, yet Muslims give charity for their dead.
The response
In Islam the reward is not a force inside the deed. It is Allah's (﷾) to give. The Prophet (ﷺ) taught that charity given for a dead mother reaches her, and that a righteous child's prayer is among the deeds that go on after death. The One who sets the account also chooses to add to it. Karma in the Pali texts has no one to do that.
Where it lands — Help for the dead fits Islam, where a Lord gives the reward. It does not fit a karma that works by itself.
Perspectives
Buddhist replies
The dead are not fed by the gift itself, but by their own joy at it. And pilgrimage and garlands help only by the faith they stir in your own heart.
The departed are helped by their own joy at the gift. Their thankful rejoicing is their own good deed, so no merit passes from one being to another.
Rebuttal
The discourse says more than that. The ghost lives on what its relatives give, and the gift itself helps the one living there. And the ghost can only rejoice because someone else gave. So its welfare still depends on another person's deed.
Pilgrimage and garlands help because they raise confidence in one's own heart, so the merit is still one's own.
Rebuttal
The promises are tied to places and offerings: heaven for dying on the way to four shrines, and welfare for garlands at one monument. And the Buddha tied a people's growth to their offerings to spirits. Those are given to others, not to your own heart.
The Islamic perspective
The Pali texts teach that each person alone reaps what he did. Yet they also teach that gifts feed the dead, and that shrines bring heaven. Islam is consistent: each soul answers for itself, and the help that reaches others comes from Allah (﷾), who gives every reward.
The Dhammapada says no one can make another pure. And King Yama says the wrongdoer alone will feel the result of his deed.
The Buddha says gifts for the dead help relatives reborn as ghosts. He promises heaven to those who die on pilgrimage to his shrines. And he ties a people's growth to their offerings to spirits.
The Qur'an teaches that each soul has what it worked for. And the Prophet (ﷺ) taught that a child's prayer, and charity given for a dead parent, reach them by Allah's (﷾) grace.