Is it permissible for a son to remain in his deceased father's house without paying rent to his siblings if he would be harmed by leaving, and is it permissible for them to rent out the house and force him out against his will?

If a person dies, their estate is transferred to their heirs by compulsion, and it becomes their property immediately upon death. The house is now owned by all the heirs, and if one of them requests its division, sale, or rental, everyone is compelled to comply. So, if your brothers ask to rent out the house, you must vacate it for rental and share the rent. Alternatively, you can offer to rent it from them at an agreed-upon rent to be paid after the house is sold. If they accept, that is good; otherwise, there is no option but to rent the house to an outsider.