No, 'fewer'. Since hamburgers are countable, it must be 'fewer'. 'Less' is fine with uncountable things like 'bread', 'salt', etc. This can be confusing because lots of native speakers make this error!

As the others have said, "fewer" is the word you want. However, if you drop the "s" from "hamburgers", you can correctly use "less": If you want to keep healthy, eat less hamburger. This works because the noun "hamburger" with no article becomes the name of a category rather than referring to specific things. It thereby becomes uncountable so you can use "less". Here are some other examples that illustrate this phenomenon: "I don't drink much coffee" "He watches too much horror" "This table is made of wood" "She plays a lot of soccer" "The recipe calls for too much onion"

No, it isn't being used correctly. The correct word to use here is "fewer." This is a tricky question, because native speakers sometimes use the wrong word. Everybody would understand what you wrote. However, it's incorrect. "Fewer" is used for countable nouns, things you can count with numbers, like "hamburgers" or "meals" or "items." "If you want to keep healthy, you have to eat fewer hamburgers." "If you want to save money, buy fewer restaurant meals." "This line is for customers with ten items or fewer." "Less" is used for uncountable nouns, like "fat" or "coffee" or "stress." "If you want to keep healthy, you have to eat less fat." "If you want to sleep better, drink less coffee." "Your job is affecting your health. Find a job with less stress."

No. Consider these two options instead: If you want to be more healthy, you have to eat less hamburgers. If you want to keep healthy, you have to keep avoiding hamburgers.