Are we obligated to relinquish our father's share of our paternal grandmother's house, given that she had relinquished her share of our father's house and died before we could relinquish ours to her?

If the father dies, his mother inherits one-sixth of his estate. She is permitted to waive her share, either gratuitously or for a consideration, and this is called "Takhāruj" (mutual settlement/disinheritance). Takhāruj is subject to the rules of sale, which stipulate that the consideration must be known and owned.

If the consideration is the heirs' relinquishment of their share and their father's share in the grandmother's house (her inheritance after her death), this is impermissible because they do not own this share during the grandmother's lifetime, and because the deceased father does not inherit from his mother if he dies before her.

However, if the consideration is the heirs' relinquishment of their father's share in the grandmother's house, which was a right the father had during her lifetime, then this consideration is valid, and you are obligated to deliver your father's share to the grandmother's heirs.