Remember, anyone offering to lend you money without a solid appraisal of the collateral is not doing so because YOU will be better protected. Lending without an appraisal just means that they are satisfied that THEY are protected (that there's enough value in the home to secure their interest).
Given the outrageous junk charges that lenders cram into loans, the appraisal is the last place to look to save money. Instead of getting a refi (which is just real estate jargon for a new loan) without an appraisal, maybe you should ask them to let you pay for an independent appraisal but cut the cost of it out of the other closing costs they charge, especially the loan application fee.
Never think for a minute that the lender is your friend -- because they are not, as millions and millions of Americans are learning to their sorrow.