Does the Army have to accept gear turned in with the labels missing?

There is no regulatory requirement directing that an item must have a label to be turned in. The label is not the item, the item is itself. If your son has the item and the Supply SGT simply wont accept it, that does not mean there is a loss. Additionally, loss of a label does not indicate damage to property. Your son needs to speak to the CDR or PBO. If he has further problems...go see the IG. Oh, and if I was him, I would definitely not sign the statement of charges. Another option is to force the Supply SGT to initiate the FLIPL and when the investigating officer shows up...just tell him the story and hand him the supposedly missing gear. Again...can't be liable for a loss or damage when there is none (still talk to the CDR tho). Look up AR 710-2 section III for OCIE responsibilities and AR 735-5 Chapter 12 for more info about loss/damage accountability.

The only thing I can think of where the labels should cause trouble (and for good reason) is with ballistic inserts for your OTV. Each insert panel is labeled with a serial number and number that corresponds to it's production run. These are important so if a product run is later found to be faulty the inserts can be pulled from issue.
Other than that a missing label shouldn't prevent turn in. Labels on pistol belts wear and tear over time as do clothing labels.

As with everything, there are standards and policies which dictate how things are processed. Labels usually contain the NSNs which are used to verify the item. Since turnin items need to match what is on the clothing record, even down to the size, the label is where this information is most likely located. Although there is often a bit of latitude, Supply must operate within the constraints established above them. If there are strict policies, it’s out of their hands.