Yes if your lease or rental agreement says that if you're late then you owe a late fee and you fail to pay that late fee then the landlord can refuse the payment for rent and then file for eviction if they want too.
However if the lease doesn't say anything about a late fee then the landlord cannot impose one and evict you for not paying it.
Late fees for renting must also be reasonable to be enforceable but yes if you fail to pay the late fee along with the rent then you can be in breach of your rental agreement for the home or apartment and then the landlord can start the eviction process or ask you to move out.
If the late fee is reasonable and you like the place then just pay the late fee if possible and avoid any other problems down the road.
Getting evicted is not good as it can prevent you from renting another place or even possibly getting approved for a mortgage or other loans.
An eviction can ruin your credit rating so it's not worth the risk of ruining your credit or having an eviction on your record for a simple late fee.