I had a visa sticker get "detouched" from my passport before by itself. It was an old visa and just fell of (was a US visa, by the way). I was out of pages, and desparate to get another visa, and had no problem whatsoever getting another visa on that page. Of course I did not mention that to the consulate that was giving me visa, and that page was otehrwise intact. And I did not really do it by myself. I blaim them for using low qaulity stickers.

But most countries stamp or sign visa so that part of teh stamp or signature is on the edge of the passport page. Even if it is not, stickers are hard to remove. Otherwise, if I am desparate and knew how to remove them - I would try. Depends on desparation.