As far as Islamic Sharia is concerned, Zakat Al-Fitr is obligatory on all Muslims who are alive when the sun sets on the night of `Eid. It is not obligatory on one who died before this time. But it becomes due on one who died after sunset of the last day of fasting, which is the night of `Eid. If in this case, the deceased did not pay Zakat Al-Fitr, then his or her relatives have to pay it from his or her wealth on his or her behalf.
In his response to this issue, Sheikh Muhammmad S. Al-Munajjid stated:
Zakat Al-Fitr is obligatory for all Muslims, males and females, old and young, as stated by Prophet Muhammad (peace and blessings be upon him).
It is only required from the living who are present at the time when it becomes due. Zakat Al-Fitr becomes due when the sun sets on the last day of Ramadan, because the Prophet (peace and blessings be upon him) called it Sadaqat Al-Fitr, and fitr (arabic for: breaking the fast of Ramadan) comes when the sun sets on the night of `Eid (i.e. the night before `Eid). It is a purification for the fasting person from any idle or obscene speech, and the fast ends when the sun sets.
Therefore, if a person dies before the time it become obligatory, he or she does not have to pay this zakat. If a person, however, lives until the time when it becomes obligatory and he or she dies before paying it, then it should be paid on his or her behalf from his or her wealth because it is still a duty that he or she owes and becomes a debt that must be paid on his or her behalf (See: Al-Majmu`, 6/84; Al-Mughni, 2/358; Al-Mawsu`ah Al-Fiqhiyyah, 23/341).
Sheikh Ibn `Uthaymeen, a prominent Saudi scholar (may Allah have mercy upon him) said, “If a person dies before the sun sets on the night before `Eid, he or she does not have to pay Zakat Al-Fitr, because he or she died before it became obligatory.”
In conclusion, the deceased or, to be more accurate, his or her inheritors are required to pay this zakat if death happened after the time when it became obligatory, which is sunset on the night before `Eid. In that case, it must be paid on his behalf.
If he or she died before the time when it became obligatory, which appears to be the case in the question asked here, then this zakat is not obligatory.
Hence, if, for instance, a person gave charity (such as food and money) on behalf of somebody whose zakat was not obligatory on before he passed on, then it is considered charity given on his behalf and not Zakat Al-Fitr.
It is proven in more than one hadith of the Prophet (peace and blessings be upon him) that charity given on behalf of the dead benefits them and the reward reaches them.