Can you keep a pet in a no-pet apartment facility with a doctor’s note stating you need pet therapy to treat mental illness?

Your landlord can most likely prevent you from having a pet in a no-pet building. While the law requires reasonable accomodations to disabled tenants, there are two issues:

1) Not all medical issues or conditions are "disabilities" for this purpose. A disability is something with a pervasive and not-readily-ameliorated impact on the ability to do basic life functions--blindness, inability to walk, etc. are common disabilities. It's not clear that depression and anxiety would necessarily constitute a disability for housing discrimination law purposes.

2) Even if your particular conditions were severe enough, and posed enough of an impediment to be considered disabilities, a landlord's obligation is to permit physical modifications to the premises to accomodate the tenant (and the tenant can be held financially liable for many of the costs), such as grab bars, wider doorways, ramps, etc. However, a modification to the rules or allowed behaviors is not necessarily the sort of accomodation a landlord must make. In this case, though, that's what you're talking about: exempting you from a rule governing the building--a rule which all other tenants are bound by and which, presumably, some may very much want to be enforced (e.g. those with allergies or afraid of animals)--not making a physical accomodation to your condition. I do not believe a landlord can be required to do this. You may be best off waiting until the expiration of your current lease and moving to a new building which will accomodate you.