What is the ruling of Islamic law concerning an oath not to enter one's father's house due to family problems with the stepmother, and is it permissible in Islamic law to stay away from one's father to avoid these problems?

It is impermissible to distance oneself from one's father or abandon him without his consent, no matter the circumstances. Rather, one must be patient with him, avoid whatever displeases him, and seek his satisfaction in compliance with Allah Almighty's command.