(1) I eat burger and pizza only.
The only things I eat are burger and pizza. I don't eat vegetables, I don't eat fruit. I have high cholesterol.
(2) I eat only burger and pizza.
Means the same as above.
(3) I only eat burger and pizza.
Technically speaking, this could mean a couple of things: A) I only eat burger and pizza. I don't do anything else in life. I only eat, and I only eat those two things. B) I only eat those two things when I do eat. But I still do other things. I take showers, and I brush my teeth, and I go to work, and I have a family. But I weigh 300 pounds.
Since (A) is impossible to do, most people would assume it means (B). In fact, (3) is often construed the same as (1) or (2). Most people don't distinguish any difference, at least in spoken English.
(4) Only I eat burger and pizza.
I am the only person who eats burger and pizza. This can mean that I'm the only one in my health-conscious family who eats burger and pizza, so it's useful sometimes.
There might be situations where (3) can take on different meanings. For example, consider the following:
I only love you.
This could mean a few different things: A) I love you and no one else but you. You're the only person I love. This would be the most common usage. B) I can only love you. I don't do anything else besides love you. I don't breathe, I don't sleep, I don't do my homework -- I just love you. C) The only thing I've ever felt toward you is love. I don't hate you or dislike you, I only feel this one emotion toward you.