I always approach a downed deer from it's back side. I take a long stick and poke it in the butt. If it's jello I get right to work dressing it out and then draging out of the woods. If it's not dead it will be tense. The only time I've ever had a deer down in front of me that wasn't dead was a small spine shot doe. She layed there playing possum until she was poked and then she started thrashing to try and get up. One through the bolier room finnished her off but I could see how it could be dangerous.
Approach with caution and you should be okay. oh. sstay away from the hoves and antlers. The both are lethal