I wanted to give an update. I was using this dulcimer during some Quarantune workshops, and while listening to the instructor(s) I was kind of idly scraping at the adhesive with my fingernail Well, by the end of the second workshop, the adhesive was just about gone! I rubbed at the remains with my finger to finish the job. The fretboard looks great now, and I'm happy to say there were no rectangles to show where the fretboard had darkened around the labels.
On the subject of adhesive, I have another dulcimer, a Black Mountain by Dave Johnston, that had residue where a strip of duct tape had been put across the back for some reason. (It came to me via eBay, so the prior history is a mystery.) I tried a bit of cooking oil on that, and also scraped with an old credit card, which was quite effective. The dulcimer needs refinishing, so I am thinking of contacting Mr. Johnston to ask for his advice.