Oy. If you really want to get technical, a burger is a type of sammie that just happens to be hot. Most of the time. Philly cheesesteak? Sammie. Panini? Sammie. Hot or cold, slap something between two slices of bread, and you have a sammie. What makes a burger is the ground patty.
Now cold.... anyone who's ever eaten cold pizza (you KNOW you have!) can attest that a cold veggie burger MIGHT be good, if done right. In my mind, most of us are stepping outside the box on this one (I am definitely a red meat and seafood kind of chick, no lie), so going cold instead of hot takes it one step further. IF done well, of course. Still, I'll admit you'd have some convincing to do.... plus, of course, THP will have to decide if cold is ok or not. But, IMHO, if you can make a good argument for cold, I can't see why not.