I recently found a dulcimer @ an antique store in Texas and had to do some repairs. I used spray can lacquer for music instruments, bought @ Rockler, on the top and back of the dulcimer and orange oil on the fretboard. Be careful when using GooGone. If you have acrylic nails, it will melt them. I have some experience with luthier projects. Also have a lot of luthier tools and can repair most hand wood instruments. 

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. 

Thank you all! And Strumelia, I appreciate the advice about careful application.  I wouldn't have thought about that danger. 

Dusty, I was concerned about the fretboard having lighter rectangles under the stickers, but that doesn't seem to be the case.  So I was thankful for that!

A word of caution- if you are using ANY kind of liquid (alcohol, oil, GooGone, etc) for this job, be very careful to not get it near the frets. If such products seep into the fret slots it can compromise the wood-to-fret grip by swelling the raw wood in the slot, and the fret may rise a bit or become loose with playing.
For any product you use to remove stickers, use a Qtip or just the tip of a rag and do not pour/drip any substance on the fretboard. Keep the substance away from the frets and don't use so much that it seeps over to the frets.

My go-to adhesive remover is (cigarette) lighter fluid, such as Zippo.  If the finish is a hard one such as lacquer, enamel or metal, lighter fluid usually dissolves old or new adhesive fairly quickly, then evaporate completely.

I'm sure the residue from the stickers will come off eventually using any of the remedies suggested here.   Just be patient.  My concern would be if the wood underneath where the stickers were is a slightly different color than the rest of the fretboard.  If that's not the case, you're probably lucky.

Once you have the residue off, you might try a light rub down with Howard Feen-n-Wax. I use it periodically on all my dulcimers, including the fretboards.

The key is to know the finish. Plain cooking oil will work. If it is rubbed oil finish any oil my be useful. Just dry it well when finished and buff it out a little.

One of those white  "kitchen erasers" should remove the residue easily.

I suggest orange peel oil.

I just got a new-to-me dulcimer (it was made by Russ Green in 1995), and someone had put stickers with numbers all along the fretboard.  I've been peeling them off but a lot of the adhesive stays on there.  What would be good to use to get it off? 

The finish on the dulcimer is shiny, but not lacquered.  I don't know what Mr. Green used on his dulcimers.  I've been using my fingernail to scrape the adhesive off, but I'm not sure how to make sure I've gotten every trace.  And the remnants all up and down the fretboard look almost worse than the stickers. 

It's such a pretty dulcimer though.  :)