I don't have any experience with HDMI but let's start with common sense.
Do you see even any sign of life from this chip at all? In particular, its the datasheet, "Section 7.7.2 Read operation" suggests that you can send I2C commands to the chip to read its ROM. Does this chip respond to your command? If you haven't tried yet, better to give it a try and see whether you get a response. If you get one, the chip is at least up and running. If there's no response, you have an electrical issue in its external supporting circuit or a soldering issue.
Other times, defective solder joints can also cause mysterious problems. How is the board assembled? Hand soldering gives inconsistent result, reflow soldering with stenciled solder paste works better but occasional defects are not uncommon, finally some PCB prototype services now offer in-house SMT assembly. QFN/BGA is especially difficult to troubleshoot as the joints cannot be inspected or probed. A microscope helps a bit, but it cannot distinguish working-but-bad-looking and non-working joints. One trick to check a suspected I/O pin fault is isolating the chip from all external circuitry, and perform a diode measurement to VCC with a multimeter. If the I/O pin's solder joint is not defective, you should see the chip's ESD protection diode drop.
Another possibility is that the external circuitry contains a mistake. Compare your schematic with the reference design and check each pin one by one. Since I have no experience I have no comment (I took a quick look at your current-setting reference resistor and all the output enable inputs, but didn't find any obvious problem).
Bringing up ASICs is often a frustrating experience, it's not working because the external circuitry is likely incorrect, but there's no feedback and all you have is a dead chip. But sooner or later it'll be alive after you have exhaustively checked everything.
P.S: Clock and reset/power sequencing are often the cause of problems, but in this case, they cannot be. You better looking at elsewhere. As the datasheet says, this chip "does not retime any data. It contains no state machines. No inputs or outputs of the device are latched or clocked. Because the PTN3363 acts as a transparent level shifter, no reset is required".