Can a HOA evict a tenant of a home that was just foreclosed on?

If this was an HOA foreclosure (not bank) then absolutely they can demand rent or evict by statute. See FL Statute 720.3085(1)(e) and (8)(a)

Yes the HOA can absolutely do this. Florida Statute §720.3085(8) provides: If the parcel is occupied by a tenant and the parcel owner is delinquent in paying any monetary obligation due to the association, the association may demand that the tenant pay to the association the future monetary obligations related to the parcel. The demand is continuing in nature, and upon demand, the tenant must continue to pay the monetary obligations until the association releases the tenant or the tenant discontinues tenancy in the parcel. A tenant who acts in good faith in response to a written demand from an association is immune from any claim from the parcel owner.

They can do this, but it has some caveats. The HOA can only demand the next rental payment due and must give you and the current owner at least 14 days notice to pay the rent. They cannot simply kick you out and must file an action for eviction in court. They cannot demand an arbitrary amount and can only demand the rent you owe your landlord. If your rent is $1500, then I suggest you pay them that amount on the first, whether you have 14 days notice or not. Fighting about it in court would be very stressful and you would be required to put the rent in the court depository.