They can either demand the rent from you to apply to the past due assessments, or lien and foreclose on the property. If they demand you turnover the rent, the landlord cannot evict you or demand you pay rent to the landlord too. If they demand the rent and you refuse, they can evict you under a revision to Fla. Stat. 720.311. While there are ways to fight this, it's expensive and doesn't make sense in most cases. As long as you're paying rent it shouldn't matter who you pay it to as long as its not more than your regular rent.
This communication is not intended to create an attorney/client relationship. It is always recommended you consult an attorney in person to discuss your case. The Law Offices of Stage & Associates practices state-wide and represents homeowners and community associations. Please visit our website at www.stagelaw.com.