The Tenants would be living there "rent free" only if they are delinquent.
If they are in "good standing", then they paid rent to some entity . . . . either the Owner/Landlord or the Mortgagee.
Normally, collected Rents are adjusted pro-rata at Closing, so that the new Owner/Landlord is compensated for pre-paid rents for periods of time following the transfer of Ownership.
If you are not confident that the Tenants will voluntarily vacate, then do as BigToe suggested, and insist that the Closing delayed until the property can be delivered free and clear of all tenants, and after any resultant damage has been repaired.
Do you h ave a skilled Conveyance Attorney representing you in this Transaction . . . . these are all routine matters in conveying Rental Property.
Dale C. Hittle of GOLDEN RULE PROPERTIES in Glover, Vermont
Where We're Always Striving To Put Together "THE FAIR DEAL"