The refinance of your mortgage loan will require that a new title insurance policy be issued to the new lender. Even if you refinance with the same mortgage lender, a new policy is still required since the prior policy insured only that specific loan. However, significant savings can be realized if you currently have an owner's title insurance policy. Known as "reissue credit", published title insurance rates can be discounted by as much as 30% for having a policy already in place.
A refinance will not affect any owner's title insurance policy you may currently have on the property. Your owner's policy is good for as long as you own the property. Paying off or refinancing a mortgage loan will have no effect on your coverage and will not incur any additional premiums to you on that policy.