He died from heart problems. Also he was not allergic to carrots.
The sound Bugs makes while munching a carrot is actually Mel Blanc munching on a carrot. He tried using celery, raw potatoes, and a lot of other things, but only a carrot would make that carrot crunching sound. According to Noel Blanc, Mel's son, Mel was not in fact allergic to carrots as was previously thought by many. People who worked in the sound studios believed this because they would see Mel spitting out the carrot after taking a bite. Mel did this because he could not speak with the carrot in his mouth and that was the only reason he spat it out.
Oh, and Foghorn Leghorn is my favorite.