I never have. Your asking if the probe that senses the Chamber temp needs cleaning. Its never stuck in the meat, why would you worry about cross contamination? If you are refering to the meat probe, I assume you'd gently with some antibacterial and warm water. I never much cleaned it either but I would guess that most do. Heck did you know there are even folks who actually clean the glass so you can see inside the chamber? Like you can see something happening in there.
I am pretty much old school, start a fire in the fire box or now plug in the electric cord, put in the meat and let it stay till done. The thermometers achieve two things IMHO. It teaches you how to maintain your temps while learning a new pit and it insures that whatever you smoke, grill, or BBQ is safe for consumption till you get used to your cooker.
Once you get used to your pit, a little experience with the meat, you'll figure out its all about patience. Low and slow and the meat will show you when its ready. BUT till then the thermometer is handy, while you gain experience.
I will say, I do use a meat thermometer with chicken still, just because its too easy to be safe.