If your goal isn't to put on loads of size then You arent going to do any harm in doing pushups every day. What you will do though is you might over train the muscles used in pushups (chest/Biceps/Triceps) Which means when you come to do those muscles on gym day you wont get the same results in terms of growth as the others.
I'm very much an advocate of allowing muscles time to heal between sessions as that is where you make all of your gains.
You will get muscle stamina from doing pushups everyday though so it all depends on what your specific goals are.
For size, I'd say no. For general muscle stamina and all round conditioning, yeah, why the hell not.