Which TV model do you have?
Be aware of the fact that very many TVs do expect video signals on their HDMI inputs, not data signals from Macs or PCs. Please consult your TV manual if PCs or Macs can definitely be connected via HDMI, or if e.g. only DVI or RGB/VGA connectors are to be used.
In case of DVI inputs for data being present, it's not a problem if an HDMI->DVI adapter, or better, cable is being used.
Many users here are overlooking this potential reason for problems, assuming that HDMI outputs must always communicate with any HDMI input, certainly also because TV manufacturers do not "advertise" such limitations loudly.
AV receivers with multiple HDMI inputs and a single HDMI video output to the TV are typically more "tolerant" and provide a sort of "converter" function. If present, feed a Mac's HDMI signal into the receiver, not the TV directly. If the receiver does not have enough HDMI inputs, consider to connect rather a video source (DVD player or similar) directly to the TV than the the Mac.
I have read that the TV needs to be powered on and switched to HDMI before the Mac is powered up with HDMI connectivity.
I doubt this will alter any existing issues mentioned here, but may be worth investigating.
If someone would invent a Gbit Ethernet to HDMI adapter cable, I could test this with the TV out in the living room.
I've checked the manual for my Sony Bravia. And it needed a VGA connection to work properly.
I attempted using a HDMI to VGA converter and everything appeared to be fine, but then when I was about 20 minutes into using my computer it started to have the same problem all over again.
I think it's the HDMI port of the Mac Mini that is messed up.
I'm going to go ahead and have Apple take a look at it seeing as I do have Apple Care Protection on it.
Thanks for all the insight, though.