Is it obligatory to complete the construction of the mosque with the money allocated for it by the deceased, or should the remaining money be returned to his heirs?

It appears that the owner of the money appointed the questioner to build the mosque during his lifetime, not after his death. Agency (wakala) is nullified by the death of the principal (muwakkil) by agreement of the jurists.

It is stated in the Fiqh Encyclopedia: "The Shafi'is defined agency as: entrusting a person with what he can do, and which accepts deputization, to another to perform it during his lifetime," whereas "guardianship (wisaya) is after death." Sheikh Ibn Uthaymin said: "The guardian (wasi) is one who is authorized to act after death, and the agent (wakil) is one who is authorized to act during life." Agency is nullified by the death of the principal or the agent by agreement of the jurists. An example of this is when Imam Ahmad was asked about a man who gave another person money to give in charity, and the giver died before the charity was distributed. He replied: "What remains of it should be returned to his heirs."

Accordingly, the questioner must return the money to its owner's heirs and inform them of the deceased's intention to build the mosque. If they complete it, that is good; otherwise, they are not obligated to do so.