Depends on what's all been put in the burger (herbs, how much seasoning, any sauces etc).
If you had a plain piece of minced beef, rolled into a burger, and ate it on a plate on its own, then you'd probably be able to tell the difference. But a burger isn't served like that. It comes in a bun, usually with a lot of salad as well as other sauces (ketchup etc). With all those combined, it'd be very hard to actually taste the difference between rare and well done on a burger.
I make my own burgers regularly and have occasionally undercooked one slightly (pinkish in the middle) and found it to taste no different, if there was a difference, it was so slight.
It isn't a night and day difference between a medium-rare steak and a well done steak.