Why won't my iPhone work after screen replacement?

Unplug the front camera ribbon cable from the logic board and see if it boots up. If it does then the front camera assembly needs to be replaced because it sounds like that ribbon cable is damaged in a way (most likely tear).

In case you want to buy the part yourself and do it yourself:

Guide to replace the part: iPhone 5c Front-Facing Camera and Sensor Cable Replacement

Where you can get the part: iPhone 5c Front Camera and Sensor Cable

If you do decide to buy it and see the yellow tabs on the part, follow instructions explained here:

Source: iPhone 5s Front Facing Camera and Sensor Cable Replacement

Post 1#:

As someone has suggested, it is probably a part used in manufacturing process. The original part doesn't have it, and it seems to be cut off. And that is exactly what I did, and everything works. There is nowhere to attach it anyway so instead of folding it and trying to fit it in, you might as well just cut it off.

Post 2#:

As for the two "tabs" at the bottom of the flex cable, I'd have to say that those are alignment tabs used during manufacturing and assembly. Looking at an OEM cable from a screen assembly, it looks like those were cut off prior to installation. The copper tape is a combination ground and heat sink for the front facing camera and should be transferred over to the new assembly. Give it a little heat to loosen the adhesive and it should easily detach from the back of the camera and the LCD shield.