Ah - the venerable push-up. They are a good exercise, but they definitely can be overdone. Over the last ten years I fell into the push-ups everyday line of thinking. The perfect exercise and all that. For awhile I was even doing 500-700 a day, but 300 was considered the minimum. On the positive side you get great endurance that carries over to running and endurance tasks. On the negative side, your strength plateaus and there is a point where more doesn't translate to strength for other movements.
Another detriment is appearance. But before I go into that, let me add a qualifier. You see your individual genetics has a lot to do with how your body responds to an exercise. What affects me in a certain way, might not affect you the same. Now push-ups can be done anytime and anyplace. They can be done in a myriad of ways. They work the body as a whole, albeit the emphasis is on chest, triceps, and shoulders. So they have a lot of good going for them.
For me, one of the things I didn't like about push-ups is how it affected my appearance. The ribcage got way too big for the rest of my proportions. I was also starting to get round shouldered. The chest muscles were too big. It was far from a balanced look. So for the past year I have de-emphasized push-ups and worked more on exercises that create strength and mobility in the shoulders and upper back. It can be more than just pull-ups. There are a lot of self-resistance exercises that can be utilized for this. My proportions are changing and I am getting away from that barrel chested, round look.
Now I still do push-ups. I do them twice a week in a lot of different varieties. They coordinate the body's strength and give a certain look to the triceps, chest and shoulders. However they are not the engine anymore. They help make up a comprehensive train.