Not paying the late fees for rent should not get you evicted because in most states the tenant and landlord laws say that you cannot evict a tenant for not paying late fees.
In some states late fees are not even allowed to be applied to late rent payments and can get the landlord into trouble.
But if you do owe late fees for late rent payments then it would still be a good idea to pay the late rent payment fees that are added onto your rent payment.
You might need your landlord to be a reference for you when you go to rent another apartment or house.
The landlord could also take you to court over the late rent payment fees and then possibly garnish your wages etc to force you to pay up the late rent payment fees.
But if the late rent payment fees are not too much that you owe then that is unlikely because it would cost the landlord money to take you to court.