I'm not aware of any out-of-the-box Ethernet control (IP or otherwise) for a TV either on the LinuxMCE or TV ends, although in theory you could add that using the DCERouter messaging system I suppose.
The only control interfaces I have seen mentioned here are RS232 and IR. Clearly the serial is the superior if for no other reason than 2-way comms and guarentee of message receipt. However, beware, there are many TVs out there that have 232 but no control interface behind it - it is exclusively for service technicians and flash upgrades, but this is usually mentioned on the product specifications. Actually look for confirmation that the port can be used for controlling/automation, etc.
I believe that there are some default templates for controlling a (very) small number of TVs out of the box. Any others, and you need to build a template to control it yourself - with the dev guys gearing up in the background, this looks like we will start seeing additional devices coming out very soon ... very exciting! Esp. with a shared/distributed, updatable template database inherited from Pluto.
Finally, for HDMI - yes, later versions of the standard include a control interface (as totallymaxed stated, CEC), I have have asked about this before as well. The issue seems twofold - 1, there needs to be Linux drivers for HDMI chipsets to even be able to access the CEC channel so that you can encode control data on it. I don't think this is even very widespread in Windoze-land much less Linux at this point, but presumably it must be coming quickly as it is one of the primary benefits of HDMI, esp when you daisy chain your AV equipment. And 2, then the LMCE software needs to add this path as a valid way of controlling equipment... I guess this latter point will only get addressed once the former has been dealt with....
