Answer:
it all depends on how you train.
if you are lifting "easy" (i.e. the typical 10-15 rep x 3 sets type of lifting) then you could easily fit in daily pushups to compliment your easy going training. this type of training will mainly tone your body. you can train everyday on this type of workout without any problems as your body will not undergo a huge amount of physical strain so it will not require much recovery time.
however if you are training hard and heavy (i.e. 4-6 rep sets where the 7th rep is failiure) then I would recommend you fit the pushups in on your off-days but don't do anything on your 2 consecutive days off as your body will need to recover. this type of training will mainly bulk you up (with the right diet) and give you bigger muscles. it is recommended to do each muscle group once a week (e.g. day1 chest, triceps, day2 back, biceps, forearm, day3 shoulders, legs, abs) to give the muscles a full week to recover.
the main thing to remember is your body needs to recover in order to grow and be ready for heavier training. which is why i wouldn't necessarily advise doing pushups every day...
there are many ways to weight train, with many different results. its hard to ask a general question like this without explaining how you are training, what your diet is like and what you are trying to achieve with your body.
if you are serious about training and achieving your goals, then you will no doubt be attending a gym regularly, in this case 3 times a week. The best thing to do is ask one of the personal trainers at your gym.... look for the personal trainer that has the body type best suited to your goals as he/she will probably have the bet advise. tell them your training regime, diet type, sleeping pattern, and the goal you are trying to achieve. Most of the serious guys in your gym will be very happy to give you advise trust me, just make sure you are comprehensive with your questions.
(sorry about bad grammar ;-)