Yes, it removes dirt, grease and old wax. But it also sucks important oils right out of the paint's finish. Use it repeatedly and you shorten the life of your paint job. Instead of dish soap, use a cleaner formulated for vehicles (available at any auto parts store).
Once you've mixed the suds, go one step further—fill a second bucket with clean rinse water. Use it to rinse the wash mitt often. That'll remove most of the road grit from the mitt to prevent scratches. When you're finished, throw the mitt in the washing machine to get it completely clean.