If I'm cleaning and need to move them I just use my hands. Doesn't seem to bother them. They just keep on doing their thing. Tongs would break their spines. Just pull easy on them if they are attached to a rock or glass.
Tuxedo urchin: good touch
Fire urchin: bad touch
Flower urchin: death touch
Seriously, unless you know that your critter is a tuxedo or pincushion urchin, use tongs (and gloves) when handling urchins. Even those which can't envenomate you can cause injuries with sharp, stabby spines.
Also worth mentioning, you want to pick them up when they're on the move, because when they park themselves, you will be both ripping tube feet and breaking spines if you just pull them off. Best bet would be to convince it to start moving off a rock and onto substrate or similar, so they're comparatively easy to lift up.
As said, though, there are a number of species that do have venomous spines, and you don't want to be picking up them if at all possible.
I have a pair of pincushions and when I was working in the LFS, I would pick urchins up almost constantly as they escaped the tank a lot and into the tanks beneath them (It was a gravity type tank). All we did was carefully take them off the glass. Never hurt unless you didn’t expect them to be there and accidentally hit them with your bare hand. As for long spine urchins, we didn’t usually handle them the way we did other urchins, usually letting them walk into what we were going to put them in.