So it's not about iridium tools or even just tools at all. You just don't want your sign changed. Why you'd never label anything with a tool is beyond me, though. Why not use a sign to label a chest with wood in it... with your axe?
And okay, I'm starting to think the better question is, "Why are people using so many signs?"
For chests? Don't put signs on them. Color code them. Red for the bloody parts you get from critters, greens for things that come from plants, black/grey for the rocks and other things taken out of the ground, and etc. This way you're not using up additional space (a valuable commodity) for something that can be accomplished in a simpler way.
In my mind signs are for fish ponds and putting places that you aren't going to be clicking all the time anyway. If it is somewhere where you're doing a lot of clicking, like amongst your chests to label them, they are in the way in some fashion. I have 7 chests surrounding my workbench that stores all the things that can be crafted with, each color coded, and I could not fit all of that stuff in a smaller quantity of chests... and if I put up signs, I couldn't access all the chests.
To be clear, I'm not against this idea. I would definitely lock any sign that I put up, and I do use signs for aesthetic purposes. I am stating that the problem you are having can be solved a better way.