Is a dump truck a commercial vehicle?

When used for commercial purposes, yes. If you're asking about this insofar as CDL requirements go, then that's a different story. If the vehicle is rated up to (but not more than) 26,000 lbs. GVWR, then it doesn't require a CDL. Examples include the Ford F350 through F750, Freightliner FL60 and FL70, etc. Now, you can load those up to 33,000 lbs. (some states allow more for intrastate and/or secondary road usage), but the moment you go above 26,000 lbs. total weight, then you're required to possess a Class B CDL.