The thing is, you are talking about a hypothetical. You don't know if he's going to finish the burger in 5 minutes. He might, or he might not. If you want to indicate confidence that he will, you could use when rather than if, but that would be kinda weird.
Your sentence is a perfectly valid sentence, it just doesn't mean what you want. You can say:
If you finished this burger in 5 minutes I will give you one more.
You can say that if the 5 minutes are up and he may or may not have finished the burger. The determiner this is an odd choice, as it suggests the burger still exists, but maybe you left him with the box closed, came back in 5 minutes to see if the burger is gone, with the instruction that if he didn't manage to eat it he had to leave it in the box. Then this burger would refer to the burger that may or may not be in the box.
You can talk about the future using the present tense, in English, but you can't talk about the future in the past (well, you can, but not just like that). You can talk about the present in the past, by referring to events that determine the state of things in the present. "Have you got dinner ready?" is a question in the present perfect, which is generally considered a way of talking about the past, and it could be used in place of "is dinner ready?".
Importantly, though, you don't need the past tense to make finish refer to a completed action, because using it in that way is denoting completion. You mean, of course that it depends whether they finish eating the burger, and that verb finish determines that the act of eating the burger must be complete.
You can talk about past-in-future in a couple of ways, and one of those doesn't involve overt future tense of the verb in question. There's will have, for the future perfect, of course, but "if you will have finished this burger" is not a natural way to say anything. You can, however, say:
I'll be back in five minutes, and if you've finished this burger I will give you another one.
The temporal location of the conditional is given by the first part of the sentence. Some people will criticise that, though even those criticisms largely fall away with a couple of extra words:
I'll be back in five minutes, and if you've finished this burger by that time, I will give you another one.
In fact, those clauses are independent. You can make two sentences of it, and it's fine:
I'll be back in five minutes. If you've finished this burger by then, I will give you another one.