Okay, I'll give it a shot.
Yes, it would be incorrect for you to write "I'll pay the next time we'll have dinner."
First, let's do away with the contractions. What you are then saying is, "I will pay the next time we will have dinner."
If it is not immediately obvious to you that this is incorrect, allow to to attempt to explain why. In your sentence, you are attempting to assure/convince somebody that the next time the two of you have dinner together in the future, you will then pay the (future) bill.
It is assumed that this is because they paid for the bill this time but that is not really important. What is important is that when you say "I will pay..." you are making a promise about a future event that was heretofore somehow in doubt.
Since they paid for this dinner this time, you did not. You want to assure them you will treat them as well as they have treated you, so you make a promise to them about something they are uncertain of in the future.
It is NOT uncertain that the two of you will have dinner together again; it IS uncertain who will pay for it and you assure them that since it was them this time, next time it will be you.
In your sentence, for there to even be the uncertainty of who will pay for it there has to be the certainty of another dinner happening. You do not say "I will pay the next time we will have dinner". You COULD say "We WILL have dinner again, my friend... and when we do, I will pay for us both." You could also say "When we next have dinner together, I pay." but neither of these mean quite what you're trying to say.
However, you surely wouldn't say "When we will next have dinner together, I pay."
Essentially, we use "We will" or "I will" when we wish to assure others of something which we feel they may be uncertain about. Another dinner was never in question; for you to have doubt about who will pay for that dinner, that dinner must already be a certainty.
The ONLY WAY you could say it and be somewhat right would be to say, "I'll pay next time; we'll have dinner." This is just manipulation, though; here, the certainty is now "next time" and you are now assuring them two things; not only that you will pay the bill the next time the two of you eat together but also that that next time will be dinner and not, I suppose, breakfast, lunch or brunch.
Which is not at all what you were trying to say, either. I give these examples so that maybe you will better understand why it is that you can not say your sentence the way you are trying to say it.
So... in recap; it is incorrect for you to say "I will (do something) the next time we will (do something)." It can never be correct for you to say "...the next time we will (do something)..." because there is no doubt about "the next time" happening. In our phrase, we assume and speak of the "next time" as if it is certain... and if it IS certain, than we wish to assure you that it is also certain that we will pay the bill. Hope this helps. (Also that it was not TOO confusing.)