This sounds simply like the wrong tenant for your building. Either they don't have the drive to take care of your property or you didn't make your expectations clear, in writing.
The last lease I had, where equipment was part of it, was very clear that repairs were my responsibility. I could replace the machinery if I wanted to, but if I disposed of the original equipment, my replacements became part of the building. If I expected to leave with my gear, the theatre's machines had to be stored in the condition I found them in... nothing unusual or out of the question about that.
In your case, I think you should make repair and maintenance issues clear in your lease contract, including the ability to inspect your property at any reasonable time. You should be comfortable that your tenant understands this. Finally, you should be comfortable that your tenant has the financial capability and desire to take care of your property. These may be the two places where you've given your relationship the ability to go wrong. A suitable deposit might also be in order, if you haven't already done that.
As for the filters... nothing says you can't apply an average maintenance "surcharge" to the lease, then have your own contractor take care of the heating system.