Thank you to everyone for the good, helpful information. The strapping tape had been in place for twenty years, I would guess, and it just broke when I grabbed a loose end and tried to peel it off. I discovered that a hair dryer would soften the tape without harming the finish and allow me to pull the tape (and those damn fibers) off, leaving only some sticky residue. Once the tape was removed, I could re-heat the residue and ball it up with my finger.
I will try cleaning the surface with mineral spirits after I have the tape removed.
My dad did a lot of projects, I believe, with some type of hand-rubbed finish. I don't believe he used a brushed on finish ever. This amounts to guesswork, but what type of hand-rubbed finish would typically be applied to a hardwood such as maple? (The workbench looks like maple to me.)
Thanks for your help.