It's easy to see by just looking at photos and videos of launches of re-used boosters that they don't. There are only a very few very small white lines, probably where engineers inspected some weld lines or other specific points on the vehicle.
Here is an example of B1061.2 launching the Crew-2 mission. Note that this is only the second launch of this booster, and it is already pretty dirty.
You can also see it on returning boosters, which are dirtier the more often they have flown, which would not be the case if they were cleaned every time.
Unfortunately, it is not that easy to see it on a normal SpaceX live stream because they only start the streams about 10 minutes before launch when fueling is already underway (starting at T-34 minutes) and the booster is already fully covered in ice. You can see it on live streams from other sources, though, here for example from NASASpaceflight.com and Stephen Marr (launch and return to port).