Dad wants to tranfer mortgaged rental property to daughter.

I am a lawyer in CT and practice in this area of the law.  I suggest that your dad hire a lawyer to sell the property to your for the price of the mortgage and have his daughter (if she can afford it) take out a mortgage to paying off the dad's mortgage so she has her own mortgage to pay and is on the deed- i.e. a simple sale of the property.  The town will get notice of the transfer when the tax conveyance form is filed and the deed is recorded on the land records.  I your dad wants to continue to be obligation on the mortgage, but give you the house, then have him hire a lawyer to draft a quit claim deed.  this will acomplish that and you will own the house and your dad will still be on the mortgage.  your dad needs to determine whether he wants to completely remove himself from the property.