What will happen if I don't back up my iphone 5c before i get the screen fixed?

I'm taking my phone in to get the screen fixed but my phone is too full to back up, if i just take it in then will it be okay or will i lose everything!

Depends on what kind of repair the shop is doing, and whether the tech knows what it is doing when he does the repair. The fixing of a screen should not wipe all the data. But there's no guarantee. Things happen.

If you can't back up everything, you will need to decide which things are most critical and which you could live without if they should get wiped. Back up the critical things and cross your fingers on the rest.

If you have pictures, music, etc on the phone, you should be able to copy those off the phone. Then back up the rest. Once the phone is repaired, move the files back on to it.