I can't say if it would damage the shoes but I think there is a strong potential. I noticed some stretching just from wearing my MXZ301s for an hour or so. I did this imediately after applying the Nikwax waterproofing so that may have helped.
I've worn my new MXZ301's around the house a little, sort of to break them in, and I can feel them stretching.
Someone in another thread made the same observation, saying it helped to put the Nikwax on them.
I did. It didn't work for me. Or, I should say that it was minimal improvement, but not enough to make that much of a significance. I would estimate that my shoes didn't even increase by 1/2 a size. I think the shoe stretcher idea is fine if you have a shoe that pretty much already fits, but you're just wanting to get a teensy bit extra room.
For me, I had bought the shoes about 1/2 size too small. After using them for a few rides, I realized I couldn't return them, so I tried the shoe stretcher idea with a wodden stretcher tool. It was a little bit better, but not much. I regretted not getting the 1/2 size bigger shoe that I needed in the first place. Oh well. Live and learn.