Wii U HDMI to DVI-D displays snow

You can not use a normal HDMI to DVI converter. The issue you're dealing with here is the HDMI-HDCP security layer preventing you from seeing the signal. HDCP is a security protocol to make sure you are only connecting HDMI to an approved HDMI licensed device.

The only way to make this setup work is to use an HDCP stripper. HDCP Strippers are considered encryption countermeasure devices, violate DMCA regulations and generally run in the $150-$400 price range.

Add your TAU model number to your question and I will add to this answer some more details on a recommendation for you.

Well normally a DVI output is compatible with a DVI output. :-D

There is nothing special about. It may differ if you have another version of the DVI input. But normally I would assume it should work.

But due to the fact that you use an adapter to connect them, it might be that your adapter has some kind of glitch or the cable too. Maybe your adapter doesn't support the correct kind of DVI-standard. To clarify this, you should post an image of your cable + adapter.

Anyway here are some ways to solve your issue:

1. Try it on another TV maybe on a computer display too. If this work, your TV input may have some issues. Otherwise continue.
2. Try it on a TV/computer display without the adapter. If this work, your Adapter might be broken/incompatible due to the different DVI-standard. Otherwise continue.
3. If nothing of the above work, it might be your cable. You can try to use another DVI which is working on your PC or on the TV of some of your friends. If this work, just replace the cable. If even this doesn't work, it really might be the console itself.

By the way, even if it's not the surest thing... might it be, that you connect the WII to your TV and you switched to the wrong input (or maybe your TV does it automatically)? The white snow symptoms look a bit like selecting a satellite input while it's not configured. But as I said, not the surest thing.