My grandmother lives in a 2 bedroom senior citizens apartment complex. Lately, she has been having trouble getting around, and my mother and aunt have been taking turns spending the night with her. One of my nieces is willing to stay with her more regularly - maybe two weeks at a time. The landlord has made it plain that no one can "stay" with her, and she has even suggested that my grandmother move out. Can the landlord tell us that we cannot stay with my grandmother in her illness?

Check the lease, there has to be one. If it states in the lease that there cannot be overnight guests, then yes, the landlord has the right to enforce that.

Maybe its time that your grandmother is in a long term care facility.