Does Internet search use more power on a 3G or 4G network?

I have an iPhone and previously had a droid. I'm on the Internet a lot, but it's like regular Internet search. It isn't videos and stuff like that. It's usually google or Facebook or something. I'm on a 3G network right now and able to stay up there for about 8 hours (including other stuff). But If I was on 4g would it use less power.

Your iPhone does not have 4G. It might show that it does, but in reality the iPhone is not capable of using 4G. You mention you had a "droid". If it was an original Droid on Verizon, then it too did not have a 4G antenna. Some modern Droids do have 4G, such as the Droid Razr, the Droid Razr Maxx, the Droid Incredible 4G LTE, etc.

If your phone were to have a 4G antenna, 4G data usage uses a lot more power than 3G, so it would likely drain the battery faster. Apple has not yet embraced many modern standards though, so your iPhone (regardless of what version it is) has no 4G connectivity. Apple (or perhaps AT&T) put the 4G logo on some iPhones to signify that it is "fast" on 3G, but in reality it is not 4G, or anything close to it.

Yes and know. It depends upon the phone and the technology.

More data means more power unless there's a change in basic technology and methods.