To eat one's heart out - to long for something unattainable
Burt Reynolds - a famous American movie star
The Bugs Bunny/Road Runner Movie is a look back on some of Bugs's best short films and presents the "real" Bugs Bunny as a highly successful movie star. By saying "Eat your heart out, Burt Reynolds." Bugs is joking that if Burt Reynolds had seen the movie he would have wished to be as successful, implying that he's a better movie star than Burt Reynolds.