I worked at a fast food place in high school. They trained us to put the condiments on the bottom bun so you taste it sooner in the tongue. I basically eat my burgers with that logic in mind now. Whatever side the door is is now down.
I always find that when I try and eat a sandwich or chicken burger upright, things (especially the meat) always falls out. So for me, the optimal way to eat it is to pick it up right side up and then flip it upside down to take a bite.
This doesn't really make a difference to the question, you can melt the cheese on top and then just put the patty on upside down so the cheese is down.