WHATS THE NAME OF FALLACY COMMITTED HERE?

What I'm seeing is a false dilemma . They are presenting only two possibilities, either you don't sell your house, or they become homeless. In most cases like this, the tenants would have the option of moving to another rental, or if you sell to an investor, continuing to rent through the new owner. 

Granted, it is possible that there's some sort of financial or logistical barrier to them moving to a new place, such as being unable to afford another security deposit, or other rentals out there are more expensive. But, that's something they'd need to articulate.

The short answer is that it is NOT your problem. The tenant is trying to “guilt trip” you. It is a tactic used for thousands of years. Do not fall for it.

If we accept that a logical fallacy is an error in reasoning that makes an argument unsound, I have to say “no fallacy” because I don’t see an argument.  I see two parties with different priorities, but I don’t see any specific argument.  There are some implied underpinnings of what might be turned into an argument, but none has been clearly articulated.

Perhaps there are errors in the content of what could become the argument (other houses may or may not be available, selling the house might mean it becomes unavailable for rent – there could be some jumping to conclusions happening) but we don’t know because the content of the argument is missing.

Perhaps there are errors in what could become the structure of the argument (how does the landlord selling the house lead to homelessness?) but again, we don’t have a structure to analyze.

Aside from the lack of an actual argument, the appeal to emotion (“not my problem”) can introduce a bias that can influence which side of the discussion one supports.

The lack of an actual argument leads me to no fallacy … although it does seem clear that the two parties are working from different vantage points with little presented to bridge those two perspectives.