If the landlord has it written in the rental agreement and rental lease paperwork that tenant agrees to provide renters insurance then yes the landlord can require you to have renters insurance to cover your personal belongings.
A landlord could get you evicted if the lease or rental agreements states you must have renters insurance and you don't have any renters insurance.
Renters insurance is pretty cheap to get. I've seen some renters insurance through insurance companies for as low as $10.00 per month when bundled with Auto Insurance or other insurance.