Should I connect the Shield pins 3, 11, 19, and 22 on my DVI connector to GND on my PCB? What consequences might I expect if I leave the shield pins floating?
Silicon Image, a producer of HD video link technologies, published a white paper (Well, Power point) with recommended DVI connections.
Key points to your question, DVI relies on many Differential Pairs for signal and ground. Not connecting them equally would break the differential routing. All Shields and ground should be wired through the entire link, from IC to connector to cable on both ends. The Chassis is the common ground reference, and you should NOT have any floating grounds. Failure to do so might not cause issues, but a common search on google shows that cheap DVI cables might also no route individual grounds, and some devices might fail to tie the Cable Shield to ground at their connector. This leads to mixed references and loss of connectivity.