In the eyes of your lender, your refinance loan is no different than the original loan you took out to buy your home. It’s a brand new loan. The lender will want to protect its investment, so yes, you will need to do another title search and secure another lender’s title insurance policy. The policy you originally bought only covers the lender for the life of that loan; when you refinance, you are essentially paying off that old loan and taking out a new one.
The new title search will look to make sure nothing has occurred since you got your original mortgage and now that would make the title unclear. Your lender wants to make sure there are no liens or judgements against the title.
You won’t need to purchase another owner’s policy when you refinance. Your title insurance issuer may offer you a discount to reissue the lender’s policy, so be sure to ask whether that is an option for you.