Moving around a weight room has caused more injuries than training has, for me.
You can trip, people leave crap strewn about, and as you mentioned it's easy to drop a plate. Shoes won't protect you from a falling 45lb but they'll probably prevent or at least greatly minimize a stubbed toe. Something stupid like that can sideline you for a week. Walking around a squat rack all woozy from heavy lifts is just begging for smacking your foot into something.
Specifically related to moving plates around though, I try to approach heavy plates like I would any serious lift. Keep the plate close to your body, grab it with two hands, and don't get lazy-tough-guy trying to hold two 45's. Take your time. Keep things organized and clean. If you're strength training you have enough time between sets to straighten up the plates.
Pro-tip for unloading deadlift plates: use a fractional or 2.5lb tiny plate as a "speed bump", rolling the inner plate onto it, which will lift the outer plates in the air. Repeat for the other side until you're just left with the interior plates (135lb if they're 45's).
It's a good question though. Similar problems develop when people finish squatting then lazily walk back to the pins and lean forward at the waist with ~300lbs on their back, trashing their lower backs. People take the lift seriously, but then revert right back to lazy-dangerous mode.
I actually find stuff like this a side (and real world) benefit to strength training: you learn how to move heavy things around. I tried to pick something up the other day and nudging it around figured it was about ~150lbs, something I would never try to lift unless I had my game face on and was in proper form.