Does someone who dies before paying all the installments on an unleased apartment—an apartment without the full payment of which one cannot enter Paradise—commit a sin, and is it permissible to lease the apartment before owning it in order to pay off the installments?

If a Muslim borrows money with the intention of repaying it, and then dies before settling his debt, and it is not repaid from his wealth, nor do his heirs or others voluntarily pay it off, then Allah will pay it on his behalf in the Hereafter. The proof for this is the Hadith (definition): "Whoever takes people's money with the intention of repaying it, Allah will repay it on his behalf; and whoever takes it with the intention of destroying it [not repaying it], Allah will destroy him."

There is no objection to renting out an apartment bought in installments, unless it is mortgaged. In that case, it is not permissible to sell it until its full price is paid, or the mortgagee approves the sale.