With Venus Fly Traps, the "mouths" close...so I'm pretty sure pitcher plants would react in a similar way. It doesn't hurt, but you shouldn't do it because each "head" or "mouth" can only open and close so many times and then it dies.
Nothing especially grievous would happen, lol. The pitchers have effectively evolved to trap insects in a shallow vat of digestive juices secreted by the plant. Here, after having been lured by the sweet scent, the bug usually drowns and is then consumed very slowly. Also, the larger pitchers create an almost vacuum-like atmosphere on the inside which hinders the escape attempts of flying insects. The lids do not close.
I need one that eats squirrels... I had a squirrel pass out on his back from overeatin in my grain bin... He was asleep out in the open.. I swear I thought I heard him snore.