No, the bank / lender cannot take a tenant's personal property. Once there is a foreclosure sale, whoever is the purchaser (these days in Florida, usually the bank buys the property because no private parties will outbid it), is the owner of the property. If the property is bought by a private party who wants it as an investment / rental party, the new buyer may be happy to leace the tenants in place, and at that point is entitled to rent payments. If the purchaser is the bank, usually the bank will evict the tenants. It often is obligated to do that, due to its commitment to the secondary market investors.
However, even if the bank becomes the new owner and evicts the tenants, it cannot keep their personal property. It has to file an Eviction Action, like any other landlord, and once it gets a Writ of Possession, if the tenant refuses to leave, the Sheriff will execute the Writ of Possesion and put the tenant's possessions on the curb.
However, before this happens, the tenants get notice it is coming, and can take all their belongings and leave, which is what makes sense for them to do.