Does doing push-ups all day burn muscle mass?

No, you lose muscle mass by over exercising without eating enough or by not eating enough to keep up your muscularity. Doing push ups all day can mess some stuff up, though, as you are stressing them out excessively. Building muscle is literally tearing your muslces, which is geneally good, as your food repairs the tears, making them bigger, stronger, more toned, and more durable. It is much better to max your self out and push for another 5 to 10 reps once or maybe twice a day, and rest or exercise something else the next day, giving your pecs a rest so that you can continue the routine the next day, and increase it as much as you can handle in a week or so. Do your self a favor and take it a little more easy if you continually work your chest muscles over the entire day, as doing 5-15 fast ones is better than trying to do 50 full nose touchers 10 times a day. Trust me, I know, for I am currently doing 2 sets of 60 full ones, plus 150 full reps on each arm by itself with 15 pound bar bells, all within about 1-2 hours, drinking a glass or two between sets. Water is the best choice by the way for a thirst quencher to hydrate your body for exercise.