Depends on the amp, Major. Welcome to the forum, by the way! It's a good question.
If you limited the input to the amp to <12V AC or DC or so, I'd say you are safe. [you should NEVER feed an instrument amp DC, but it can happen accidentally...] Most of our stuff is 9V...sometimes 18V. If there is an input cap on the amp (should be on solid state, some tube amps have them, some don't)...it may be worth checking its voltage rating if you can. After that, you may find an opamp or other circuitry that could be somewhat sensitive to what you feed it. I've never blown an amp this way (burst box ha ha), but I've probably damaged my speakers once or twice by feeding it a signal with WAY too much bass...
Limited to 9V or 12V, I'd say no, you won't hurt anything. Especially if you ALWAYS make sure you have a nicely rated cap on the output of the breadboard, rated for more than the power supply of what you're working on [to block DC]. Too much bass with a cranked amp caveat aside

I always start out with the amp volume very very low, and adjust accordingly...haven't run into any problems by doing things this way. I use a small practice amp for BB first, and if the circuit is worth it I move to my larger amp(s) after I know it's working.
Enjoy the forum!