How much battery drain do push notifications cause on Android 4.1?
Push notifications are designed to be the solution to battery draining apps that constantly monitor for data that has changed locally or on a server. Because of this they are very battery efficient. My phone typically will get 20 notifications in a few hours with less than 1% total battery use.

If your having huge battery drain problems, your phone is probably not sleeping properly which is usually due to an app that is preventing the phone from sleeping. I might be able to help you resolve this With some more information, such as Phone, Android version (or ROM if your rooted) and a screenshot of your battery screen after your phone has been running for a few hours. I will update my answer accordingly.