Yes, breathe while doing pushups. Conventional wisdom, as with most muscular effort, is to exhale as you're expending your effort (pushing-up) and inhaling as you move back (lowering yourself to right above the floor), but I personally find that, if you don't think about it too much, your breathing usually takes care of itself. It's worth noting that, probably due to the muscles involved, [some people have trouble breathing normally during pushups][1], but you should try to at least breathe some.

Holding your breath during a heavy lift is generally known as [the Valsalva maneuver][2] and it has contraindications:

> It's worth emphasizing that the Valsalva maneuver is only for short-duration, high-exertion efforts. The same technique that provided a core of strength for your PR back squat can become a serious headache—literally—when you apply it to a run-of-the-mill bench press.

> Many beginners—and a few experienced lifters—stop breathing during repetitive, low-intensity lifts, either because they think it'll make them stronger, or because they just plain forget. An extended Valsalva maneuver like this can cause a dangerous spike in blood pressure, bursting blood vessels in your eyes and forehead, causing headaches and temporary vision disturbances.


  [1]: http://fitness.stackexchange.com/questions/4914/difficulty-breathing-in-push-up-position?rq=1
  [2]: http://www.bodybuilding.com/fun/know-when-to-hold-it-how-to-breathe-while-lifting.html