If my dad passed without a Will but his the house has a mortgage on it, do I have to get another mortgage to keep it?

First of all I assume that you have lawfully inherited this house. That being the case, the general rule is that  person who inherits a property becomes responsible for paying the mortgage on it. Further, unless the mortgage has an "acceleration clause" that is triggered by the death of the mortgagor, the lender is typically required to continue to accept monthly payments until the mortgage note maturity date. The fact is that this benefits the person who is inheriting since they do not have to go through an application process and qualify for the mortgage on their own.
 
Note: An acceleration clause is a provision in a mortgage that allows a lender to demand immediate payment in full of the remaining mortgage balance if certain events occur, such as the sale of the property,  late payments, or the refinance of the mortgage loan. Generally, however, it is rare to find such clauses that provides for acceleration upon the mortgagor's death.
 
The fact is that, so long as you keep making all mortgage payments on time, there should be no problem with the lender. Yet, obviously you can always get a new loan in your name.