The risk with leaving the strap on the camera while on the tripod comes down to:
Risk of unintentional snagging: If you, another person, or maybe pet, were moving close by the camera, it's possible to snag the strap while moving, and disturbing the shot (at best), or pulling the the rig and possibly dropping the camera to the ground.
Wind catching strap and causing blur: This risk is greatly mitigated by proper tripod technique. But under high enough wind conditions, straps (especially manufcaturer straps that are broad) will be moved by the wind, possibly enough to slightly move or vibrate the camera, creating an image with suboptimal clarity. The chances for this go up if you do any of the following:
Extending the tripod's center colum increases the amount of vibration that is transferred to the camera.
Using a tripod and/or mount (ballhead) too small for your camera/lens makes the camera more susceptible to vibrations from the tripod (i.e., something hitting the tripod legs).
Not hanging weight from the tripod (such as a bag of rocks or sand, or your camera bag) below the apex of the tripod: mass dampens vibrations. A lightweight tripod with weight hanging below it acts very much like an old school heavy, wooden-legged tripod, with tons of stability.
Now, besides convenience, the only real benefit to leaving the strap on the camera is if you intentionally and consciously use it as a safety mechanism, to prevent an accidental gust of wind or other person from knocking the tripod over and dumping your camera to the ground. Again, this requires you to be very conscious of the strap, keeping it around your neck/arm/wrist, and being careful not to accidentally move and pull the camera with you.
In my opinion, especially because I often take long exposures, or bracketed shots that can take awhile to expose, I do not leave a strap on my camera. I have moved to a Black Rapid -style sling mount connected to a quick release plate on my camera. Thus, in order to mount the camera on a tripod, I must remove the strap QR plate, thus removing the strap.