You will change your insurance from a homeowner's policy to a landlord's policy. You may add a property management company as an additional insured. Your insurance company will send out a new policy that will be input into a computer by a clerk in the insurance department at the loan servicer. As long as the coverage is at least the loan amount, the insurance department does not care. There is no connection between the insurance department and any other department.
I have never done an FHA loan, so I have no experience with the documentation FHA requires. In many other cases you sign a rider that says you intend to occupy the property. That satisfies Fannie, Freddie, or whoever else owns the loan. Realistically, this happens all the time. People lose jobs, get transferred, or move home to deal with a family illness. They rent the property so the payments can be made.
Would I do this the week after I closed on the refi? No, because that might cause someone looking at the paperwork during the transfer from the bank or broker to the loan purchasor to question your intent at the time you signed the intent form. Would I do it 6 months after closing? Yep, because I intended to live in the property at the time I refinanced, and my circumstances changed.
Get out your loan documents. Look for the owner-occupancy form you likely signed. Read it carefully. Then decide if you are comfortable moving. If you are uncomfortable, consult an attorney that is familiar with these riders. Or just wait out the year.