Is the buyer obligated to pay the overdue installments on the furniture he purchased in installments, after he authorized a relative to pay them and the latter failed to do so, knowing that the relative claims that what he paid is the actual value of the furniture and did not return the amounts he did not pay?

If you purchased furniture by mutual consent, you are obligated to pay its agreed-upon value. You must fulfill the contract, as Allah the Almighty said: "O you who have believed, fulfill [all] contracts." And if you appointed your relative to pay the installments and gave him the money, but he did not do so, then he must return what you paid to him.