Assuming the input is detected and indicated on character LCD, the issue is probably with IT6616 or its connection to HDMI socket. This can be verified by checking (HSYNC, VSYNC, PCLK and DE) traces between FPGA and IT6613E. If there is continuity between the chips and none of the connections is e.g. shorted to GND, then at least FPGA is providing the minimum that is required for generation of valid sync at output (If you have a scope and steady hand, you could also probe those signals).
If they are ok, I’d next measure output voltages of IT6613E supply regulators U15 and U16 (both should be 1.8V). If they’re fine, then the problem could be e.g. bad connection between IT6613 and HDMI socket.