Just make sure your tenants know they are liable for any damages or disturbances he causes.

I was under the impression that anytime you disturb lead based paint, which would include ripping out drywall with lead based paint on it, that you had to have a special contractor come in that is licensed in this area to remove it and it can really destroy a budget.

Is he disturbing anyone else in the building?

On the other hand it's disturbing to know there could be that much dishonesty going on.

While I'm not a lawyer and certainly not intimately familiar with the law in MA, the information that I've seen posted in several places says that you do not have the right to cause a disturbance outside your home.

Also, I learned that an animal that misbehaves, disturbs or toilets can be removed from a public place even if it is an actual "service" animal.  

It would be easier to evict them for unauthorized pets or disturbing the neighbors than for lying about their social media account. 

Is the unit quiet enough in that closet that it doesn't disturb tenants?

Call the cops if anyone is "hanging out" on your property or causing a disturbance (noise).

I understand people have issues with disturbances caused by short term tenants however long term tenants can be a nuisance too; and they ain't leaving.