No. A permit is not required to close the loan. A permit will be required at the first draw after closing.
With that said, you certainly want to make sure that you don’t anticipate any issues in obtaining the permit prior to closing. Any sort of delay in obtaining the permit will delay the first draw.
There are some cases where a permit may not be required for your project. If that is the case, the lender will want to obtain verification from the city or municipality that states no permit is required for the scope of work being performed. Trying to do a project without a permit when one is required by the municipality will not work with a lender construction loan.