I've had a lot of experience as a consultant doing loan reviews. Most banks I've worked in require the tax returns to be signed with an original signature, i.e., take a copy of the return that was filed, then sign it again when it is submitted to the bank. Some banks used a stamp stating, "I certify this is a true copy of the original" or similar language for the signature. This is done for the reasons stated by Okie Banker.
There were plenty of banks that did not get signed tax returns, however, most did get signed returns. Regulators occasionally criticize the unsigned returns, but not a major criticism if that's all that's wrong.
Where I've seen banks go from not getting a signature to requiring it, it has not been a problem other than some initial grumbling from lenders. It really is not much work for the borrowers to sign the returns and they want the loan, so they are willing to do it. Once they have the loan proceeds, it's much harder to get the borrowers to come in for any reason.
My recommendation would be to begin to have the returns signed going forward for all new loans and renewals and to use a stamp with appropriate wording stating they certify that the return is a true copy of the original filed return.