All perfect praise be to Allah, The Lord of the Worlds. I testify that there is none worthy of worship except Allah, and that Muhammad is His slave and Messenger.
The person asked about in the question should have stated in his will that his heirs must pay the expiation for a broken oath that was due upon him. Al-Buhooti said in Kashshaaf Al-Qinaa’: “The Wasiyyah (will) is only obligatory on someone who owes a debt that lacks proof or keeps a deposit with him without proof, or someone upon whom Zakaah, Hajj, expiation, or a vow was due. He is required to make a will to the effect of fulfilling such obligations on his behalf, because it is obligatory to render trusts back to their owners and to fulfill the obligations.” [End of quote]
Expiation for a broken oath should be paid from the inheritance of the deceased before dividing it among the heirs. Shaykh Zakariyya Al-Ansaari of the Shaafi‘i School of Fiqh, said in Asna Al-Mataalib: “If a free person died while expiation for a broken oath was due on him, then it is considered a debt that is due to Allah, The Exalted. Since the rights due to Allah take precedence over the rights due to humans, then it is paid from the deceased’s estate before the shares of heirs are distributed. This is to be done whether the deceased stated it in his will or not, given the statement that: ‘the debt to Allah is worthier of being paid.’” [End of quote]
Scholars held different views regarding the issue of fasting on behalf of a deceased person. Those who held the validity of doing this said that it is permissible if the deceased was able to make up for the fasts before his death but did not fast. In such a case, it is recommended that his heir fasts on his behalf. This issue was subject to extensive research that is too lengthy to explain in detail here.
If the deceased was liable to payment of expiation for a broken oath, fasting, or other rights that are due to Allah, The Exalted, but he acted neglectfully towards them, then we cannot decisively confirm whether Allah, The Exalted, will forgive him or not. Rather, he is under the Will of Allah, The Exalted, Who may forgive him if He so wills or punish him if He so wills.
Shaykh Al-Islam Ibn Taymiyyah said in Al-Fataawa Al-Kubra: “When a slave wrongs himself concerning what is between him and his Lord; this type of wronging can be forgiven without the need to get the people’s consent. That is to say, Allah, The Exalted, may punish this person who wronged himself if He wills or forgive him if He wills.” [End of quote]
Allah Knows best.