This will ultimately depend on the bank. Each bank or lender will have their own rules on application fee refunds. Most banks charge an ‘application fee’ rather than an appraisal fee so that they do not have to give a refund. You are literally paying a fee to apply for a loan. This does not mean you will get a loan, this is simply a cost you are paying to apply. Because it is structured this way, the bank has no requirement to refund the money under any circumstance, whether an appraisal was done or not, or whether the loan is moving forward or not.
Despite this, most of the time a bank will have some type of refund policy in place. If you have a good banking relationship with this bank they will be more likely to give you some type of refund if it was their error.
If the loan is not moving forward because the appraised value was too low, or because you have decided not to move forward don’t expect a refund. Appraisers are not employees of the bank or lender, therefore if they do an appraisal they require payment for their work. If the bank refunds your application fee because the home appraised too low, they still have to pay the appraiser. Multiply this over a few thousand loan application a month and you can see how a bank can’t afford to refund fees due to low appraisals.
While it certainly isn’t the answer you were probably looking for, if you put yourself in the banks shoes you can see how they would be unable to offer refunds for low appraisals. Given the current market state it would put a bank out of business if they had to pay 3 or 4 hundred dollars every time an appraisal came in lower than anticipated.