The idea is to build a server and run virtual operating systems compatible with touch screen monitors. Server in one location with four displays in different rooms. The idea is to only have the display as a hardware element in the room hardwired to the server which is in a separate location. The goal is to have the ability to let the kids have their own computers without them having access to any hardware outside of the display. They would in essence have their own VM but the trick is to eliminate mouse and keyboard. I believe this setup would be possible but my concern would be limitations on the wiring. Is there a length limit for HDMI cables?
Yes an no... from Wikipedia's HDMI entry
Although no maximum length for an HDMI cable is specified, signal attenuation (dependent on the cable's construction quality and conducting materials) limits usable lengths in practice.
A cable of about 5 meters (16 ft) can be manufactured to Category 1 specifications easily and inexpensively by using 28 AWG (0.081 mm²) conductors. With better quality construction and materials, including 24 AWG (0.205 mm²) conductors, an HDMI cable can reach lengths of up to 15 meters (49 ft).
I assume that USB distance is not an issue? You'll need USB for your touchscreen and other input devices. In that case, you might consider using a USB video adapter next to the monitor, instead of running HDMI all the way to the host.
The big problem with these adapters is performance. But I'm guessing the apps you intend to run will not be high-performance, since they're running in a VM.
According to http://www.hdmi.org/learningcenter/kb.aspx#44 there is no maximum cable length:
HDMI specifies the required performance of a cable but does not specify a maximum cable length