That is an odd one, though I vaguely remember reading a similar thread here once before, which may or may not have come to any solution.
To start with, wireless or wired internet? Assuming wireless, can you connect it to the hub via an ethernet cable and verify that it doesn't drop out? (Just to rule out an extremely unlikely software glitch.)
More probable seems like either some sort of interference and/or ground-loop issue (those can cause some pretty funny behavior), or maybe some kind of a hardware problem in the computer that's only shorting out when an external display is plugged in.
A few more things to try to narrow it down:
- Try plugging it into the TV with the TV off.
- Try connecting an HDMI cable and adapter that ISN'T connected to anything at the other end.
- Try it with the computer completely disconnected from power or anything else external--just the TV and the internal battery.
- Try it with the computer connected to both the TV and its power brick, but with the brick plugged into an outlet on the same power strip as the TV.
