IT IS A MITZVAH TO FULFILL THE REQUEST OF THE DECEASED [Shelichus:posthumous]
Gemara
(Mishnah - R. Meir): If Levi gave money to David for Levi's daughter (for her dowry or to buy a field for her), and she says 'I trust my husband (give the money to him)', David should do like he was instructed;
R. Yosi says, even if he bought a field for her and she wanted to sell it, she could. It is as if it is already sold! (Therefore, we heed her.)
Question: What is the source (in the Mishnayos) of the following Tosefta?
If one said 'give a Shekel to my children each week', and they need a Sela, we give to them a Sela. If he said 'give to them only a Shekel', we give to them only a Shekel;
If he said 'if they die, others should inherit in place of them', whether or not he said 'only', we give to them only a Shekel.
Answer (Ilfa): That is like R. Meir, who says (in our Mishnah) that it is a Mitzvah to fulfill the words of the deceased.
(Rav Chisda): The Halachah is, whether or not he said 'only', we give them all they need.
Question: The Halachah follows R. Meir, who says that it is a Mitzvah to fulfill the words of the deceased!
Answer: That is in other cases. Here, the deceased wants his children to have their full needs. He said to give to them only a Shekel, merely so that they will be frugal.
Rishonim
The Rif and Rosh (6:22) bring the Gemara verbatim.
Ran (DH v'Re'uyin): He said 'only a Shekel' because he did not expect the price of food to rise.
Rambam (Hilchos Zechiyah 11:23): If a Shechiv Mera (a sick person who fears lest he die) said 'give a Shekel to my children each week' or 'give to them only a Shekel' and they need a full Sela, we give to them all they need. He did not intend to starve them. He merely wanted to encourage them not to spend more than necessary.
Rambam (12:6): If a Shechiv Mera (Yosef) said 'my property is to you (Ploni), and after him to Almoni', and Ploni was fit to inherit Yosef:
If he explicitly said 'I do not give to you for an inheritance, which cannot be interrupted, rather, for a gift, and I interrupted it', Almoni gets what Ploni left over. Therefore, if he gave coins to a third party, or said 'give a Shekel to my children each week', not for an inheritance, and after them to Almoni, we give to them only a Shekel, even if they need more.
Ra'avad: The Rambam said this to answer the question about 'give a Shekel to my children each week.' His answer is poor. I have a better answer.
Magid Mishneh: Ge'onim say that even a gift to an heir becomes an inheritance. Many ask, if so, when he said 'give a Shekel to my children each week, and if they die, others should inherit in place of them', why do the others receive? Inheritance cannot be interrupted! The Rambam answers that if he explicitly says 'it is not an inheritance' it can be interrupted.
Beis Yosef (CM 248 DH u'Mah she'Chosav v'Chegon): The Ran answers that because Yakov never gave all the property to his children, only a Shekel per week, the stipulation helps for the rest. Others says that there the property went to a third party, not to Yakov's children.
Drishah (248:1 DH v'Ika): The Rambam says that the property was given to a third party and he explicitly said 'it is not an inheritance.' This connotes that one of these alone is not enough.
Rosh (22): Since we hold that it is a Mitzvah to fulfill the words of the deceased, why did Chachamim need to enact a gift of a Shechiv Mera? R. Tam says that it is a Mitzvah to fulfill the words of the deceased only when the property was given to a third party. It also applies to a slave (her master commanded to appease her, through freedom if necessary - Gitin 40a). She is Muchzekes in herself, which is like being given to a third party.
Poskim
Shulchan Aruch (CM 253:17): If (Yakov,) a Shechiv Mera said 'give a Shekel to my children each week' or 'give to them only a Shekel' and they need a full Sela, we give to them all they need. If he said 'if they die, others should inherit in place of them', we give to them only a Shekel. See 248:1.
Beis Yosef (DH v'Amai): Rashi explains that when he said 'if they die, others should inherit in place of them', this refers to dying without children. This is because when David gives 'to you (Moshe) and after you, to Ploni', if Moshe is fit to inherit David, Ploni gets nothing, for one cannot interrupt inheritance. If Yakov's children die without children, the stipulation takes effect. It seems that this is when his children die without any heirs. Alternatively, since their heirs are Yakov's brothers, who inherit through Yakov, therefore Yakov can stipulate to stop this. One cannot interrupt inheritance when the children have children or brothers whose inheritance does not depend on Yakov. I gave other answers in Siman 248 (see below).
Beis Yosef (DH v'Ika): We hold that 'Ploni should inherit' does not help if Ploni is not fit to inherit! We must say that the Beraisa is imprecise. Really, he said 'others will receive'. Why didn't Rashi explain this? Rather, we must say that the others are fit to inherit. However, there are other heirs, and Yakov's stipulation causes the one(s) he specified to inherit alone.
Rebuttal (Shach 23 and Taz): The others are not fit to inherit. Yakov's initial words 'give a Shekel to my sons...', help for the others, even though later he calls it inheritance, because he said it all (without pausing) Toch Kedai Dibur. If the stipulation works when Yakov's children died childless, the Gemara should have answered like this in Bava Basra (129b)! Rashi said that the children have no children, for then Yakov is concerned lest his children not be thrifty, since they have no children to whom the rest will pass.
Defense (Ketzos ha'Choshen 9): The initial words 'give...' apply to Yakov's sons. They do not help for others, to whom he said only 'inherit'. Since he said 'after', his stipulation takes effect only after death, therefore it cannot help if there are heirs, for inheritance takes effect at the moment of death.
Prishah (33): If the others are fit to inherit, Yakov did not need to stipulate! Indeed, he stipulated only so that his children will not receive more than a Shekel each week.
SMA (38): What they need above a Shekel they receive from Tzedakah. We do not heed one who says 'do not bury me from my property'. There, the property is his (and after death he cannot change this). He cannot command not to use it in order to benefit his children. Here, the property does not belong to his children, rather, to Yakov. He could give it to anyone he wants, and force his children to beg!
Shulchan Aruch (248:1): If a Shechiv Mera (Yosef) said 'my property is to Ploni, and after him to Almoni', Almoni gets only what Ploni leaves over. If Ploni was fit to inherit Yosef, e.g. he was one of Yosef's sons, Almoni does not get anything.
Rema: This is even if Almoni is also fit to inherit Yosef. If Almoni was Makdish the property, nothing happens.
Shulchan Aruch (ibid.): This is because any expression of 'gift' to an heir is like inheritance, and inheritance cannot be interrupted, even if he said 'after him to Almoni.' If a healthy person gave a regular gift like this, Almoni gets only what Ploni leaves over, even if Ploni was fit to inherit Yosef.