eslHQ - ...time...

First of all, note the spelling of minutes and hamburger.

It's difficult to assess the feasibility of these sentences (or understand their meaning) without knowing the context. Context determines meaning, and grammar is chosen to express that meaning. For example in (1) I suspect you mean :

A : I've got too many sandwiches here. Do you want one?
B : No thanks. I'm going to McBoggles with John to have a hamburger in ten minutes.

ie - you're expressing your intention. You therefore need "Be going to" not "will" and (probably) "have" rather than "eat": I'm going to have a hamburger in ten minutes.

The only context I can think of (changing it a bit) for your first example is one of those horrific competitions where people have to eat vast quantities of food in as short a time as possible. Before the competition one of the competitors is boasting : You'll see - I'll eat twenty hamburgers in five minutes.

Here "will" is fine because he's making a prediction.
Notice that there's a difference in meaning in the time phrase between the two examples :

I'm going to have lunch in ten minutes. ie - it's now 1.05,and I'm going to have lunch at 1.15, ten minutes from now.
I'll eat twenty hamburgers in five minutes. = it will take me five minutes to eat twenty hamburgers or I'll eat them during a period of five minutes .

Example 2 makes no sense really. Compare it with eg :
I swim for 50 minutes every day.
Here you're describing a regular, repeated action (so the present simple is used) and the frequency is stated - every day. My background knowledge tells me that swimming is good exercise and exercise keeps you healthy - so I understand that you're telling me how you keep healthy. But why ever should you tell me that you eat hamburgers for ten minutes every day - ??
The only thing that I can think of is that you really meant - It takes me ten minutes to eat a hamburger , ie I can eat a hamburger in ten minutes -???

The final example is incorrect grammatically as well as not meaning anything.