You can call a restaurant a "place" in casual conversation:
Have you been to that new pizza place up on 83rd Street?
Calling a restaurant a "place" is most common for shops that aren't sit-down restaurants with waiters, but also aren't big chain fast food restaurants. For these kinds of "places", the word "restaurant" can seem too formal in spoken conversation.
You can use the word "place" by itself as well, without naming the type of restaurant. For example:
I really like this place.
"Burger" is a casual term for a hamburger.