Is a capable father sinful if he threatens to evict his son from an apartment he owns should the son fail to pay the rent on time, even though the father is aware of his son's struggling business and his need to borrow money to pay the rent?

There is no harm in your father asking you for rent, as he is not obligated to financially support you, and housing is part of financial support. It is better for him to be considerate of you and not threaten you with expulsion. You must be patient with him and be careful not to be disobedient to him. If you consider moving out to avoid discord, there is no harm in that.