You would never not shower after working out, and in the same way, you should always clean your machine (at least superficially) after using it. It is very normal to sweat extensively while you are running on the treadmill, and the excessive sweat can accumulate on the running belt. This will make the belt slippery and wet, which is potentially dangerous for the next person. The running belt needs to be cleaned with a dry cloth after every use, especially if the person using it sweats a lot while exercising. Even when used by family members, it is hygienic to clean the treadmill after someone has used and sweated on it – both the running belt and the handle.
If possible, the treadmill needs to be used somewhere with fresh air available; exercising and sweating in an enclosed location with increase the chances of unwanted body odor around the treadmill. Use a fan, open a window, or make sure that the AC is on while your're getting your cardio on.