First, when your toilet tank is getting pretty full, you will start noticing the stink inside, since it's past the vent pipe to the roof. You don't want that smell seeping in for long. Also, at that point, the toilet will start to "burp". That is, when you flush, a bubble will come up and burst. Since the level inside is past the vent pipe, the air inside the tank has to come out when you flush and put more "stuff" in. Drain immediately if that's the case since there probably isn't much more room in there for more than a couple of more trips to the bathroom.
Second is your grey tank. Some are built differently, but if your grey tank is an isolated tank with no overflow protection to the black (toilet) tank, when it gets full, the water will start to accumulate at the bottom of your tub, since it's lower than your sinks. That will be time to drain. If there is some kind of overflow protection (yes, it does exist, as disgusting as it sounds) to your black and you fill it up (essentially BOTH will be filled), that water at the bottom of your tub could and will be a mix of both grey and black tank water. UGH.
Overfilling your water tank isn't much of an issue. Should be overflow protection, but even if not, no damage will occur since it'll be like overfilling your gas tank - it'll spit out at you.