It depends on the drymount tissue that you choose. I didn't read the other responses here but I am willing to bet that drymounting got a bad rap by some respondents. Personally, I think that is unfortunate because, when done right, drymounting is a very good way to go. Sure, the prints will end up in the fine print collection of a museum, you will not want to drymount. However, let's be honest and ask ourselves what our prints will really be used for. I like to drymount all my prints. They lay flatter and simply look better. It also offers a measure of physical protection. Remember that Ansel Adams was fully aware of both the pros and cons of drymounting and yet he preferred to drymount all his prints.
As far as the adhesive goes, most if not all drymount tissues will not harm your prints. I have prints that I drymounted decades ago that look like they did the day I mounted them. Check out Wilhelm Research if you must have the most comprehensive answers to your question. You can even download Wilhelm's entire book on the subject in PDF format! Now that's what I call generous and he is to be commended sharing it with everyone for free. I downloaded it and put it on CD in case it isn't available for download in the future.