1 cup isn't very much at all. No, sugar will not generally lead to an alcohol burn. Chances are this is due to the temperature being too high during fermentation.
What strain of yeast did you use and what temperature did the beer ferment at? It can be tough to maintain appropriate temperatures this time of year without something like a swamp cooler or dedicated fermentation chamber, but most ale yeasts are happy in the mid to upper 60s. And since fermentation generates heat (over 5 degrees isn't unheard of), that means you should have them someplace cooler than that during active fermentation. A 60-65 degree room is ideal. The "burn" is fusel alcohols that are produced at higher temperatures. It's a lot easier for the temperature to get out of your control with a stronger beer like an imperial IPA.