Can denatured GFP show fluorescence?
According to PDB (protein data bank — a repository for protein structures) the chromophore must be protected from interaction with water molecules to fluoresce.


  The chromophore is found right in the middle of the [protein], totally shielded from the surrounding environment. This shielding is essential for the fluorescence. The jostling water molecules would normally rob the chromophore of its energy once it absorbs a photon. But inside the protein, it is protected, releasing the energy instead as a slightly less energetic photon of light. 


This means that denaturing the protein will result in quenching of the fluorescence1, so you are unlikely to see significant signal on a denaturing gel.  In fact, as WYSIWYG has pointed out, fluorescence has been used as a marker of renaturation of GFP1.

Reference:
1: R.Y. Tsien (1998) The green fluorescent protein. Annual Review of Biochemistry 67, 509-544.