Insteed of refinancing a 10.5% mortgage on $40,000. Suppose you just took
Out a loan at 5% or anything below the 10.5% and paid off the mortgage

Taking out a loan at 5% or anything below the 10.5% and paying off the mortgage on $40,000 is the definition of 'refinancing'. Your concern may the costs of getting this new loan, or 'closing costs', points, fees associated with the new loan, etc.

If its not a mortgage, you lose the tax benefits.