It's less dust and more ash from the wood fire. It shouldn't be much of a concern for a few cooks, but it really depends on how clean you wanna keep your grill. A bit of soot isn't going to hurt the painted walls, but it can build up inside the fire deflector underneath the "flavorizer bar". If you remove that, you'll see that big honkin' silver thing in the center. Take that out and dump it out after a couple of short cooks or a long cook.
While it's off, you can also check your fire box too. There are two wingnut screws you can remove and then carefully remove the fire pot so you can dump out any ash there too. I usually leave unburned or partially-burned wood pieces. They'll burn through once you light the grill up the next time.