If you need to replace the glass on the unit, i can imagine it's because the unit was victim to accidental damage?
If so, the reason why the problem only arises when the display is seated, is because the enclosure is bent out of shape, and the new LCD panel can't handle the pressure (so to speak).
I've seen this on original screens and 3rd party displays, but mostly on units with a bent enclosure, and it does not have to be a lot. The 5C display does not have a whole lot of flexibility, like the 5 and 5S, because of the extra plastic framing, i'm afraid.