Krista, if your countertop was glued in with a lot of liquid nails or something like that, getting it out could damage the wall or backsplash. Maybe that's what they mean. In general the backsplash wall will need at least some patch repair when you pull off the counter.
But it's so worth it. When you are on a budget, and looking at replacing a really hideous Formica, I really love wood. It's inexpensive and great looking. You just need to treat it with a wood oil. And if it's going over a dishwasher, apply foil backing to that spot.
Some of the cabinets are installed around the Formica so they would have to be removed and put back, assuming they don't get damaged in the process. The amount of Formica in my kitchen is sort of astounding, so replacing it will change the whole feel.
The strange thing about butcher block counters is that they seem very popular on design blogs and I know they are sold inexpensively at Ikea, but the two contractors I've met with seem to think wood is a specialty item and didn't seem crazy about working with it. I really dislike granite and quartz is pricier than what I'm looking for right now. If I were doing stone at all, I would choose marble (which I've been warned against).