Well I've used oil to remove paper and the sticky labels from boxes ? I use clear coat on my paper boxes to seal them using very thin coats to build up a finish .
I'm guessing you mean an oil based finish. Like lacquer or polyurethane. I'd try a test area first to make sure your not going to smear the ink. But ya. People have used oil based finishes to preserve paper for years. Most will turn yellow.
I doubt there's any point in using Tung or Tru oil over paper covered boxes,i've never done it,so just surmising,i'd expect it would tend to release the paper cover at best,and may even get messy,probably better served with an acrylic clear coat/sealer,a good 2 pack clear would probably 'stiffen' the sound board as well,but,just my thoughts,no experiance to draw on
I'd go with poly ,lacquer or shellac on the paper covering.
I've been using spray Shellac to help protect the box paper. I've heard it yellows over time but have not seen anything (yet). My new favorite is clear acrylic spray. It goes over any material or finish and dries hard and fast. You can use as many coats as needed. Poly was taking too long to dry and the dust nibs kept it from getting a smooth finish.
I have in the past just covered my labels with diluted PVA craft glue. 5/1 water. Dries well, seals and clear finish. Final coat of water based satin varnish does the trick for me. Cheers