Will this cause my computer to use more energy or power?

That won't happen. The cpu won't use power but the monitor itself will require more power but that's what the electricity's for so no worries. Everything will work just fine. Hope this helped =)

No, it makes no difference to the power the computer uses. The giant HDTV will of course use much more power than the 23" monitor.

You don't have anything to worry about if your concern is the computers energy or power consumption. Remember that the new or old set up are both self powered. This means they do not use your computer for energy or power source. So no it will not effect your computers energy consumption but can effect your homes overall kilowatt usage or consumption very minimally. I say enjoy it! Sounds awesome to me. I did this back when plasmas first came out but not with an HDMI that's even better.

Just read the power rating on the back sticker of each monitor. The monitor won't draw more or less from the PC, (the cables have specific impedance specs for compatibility) but may draw more or less from the wall. The brighter your screen, or the bigger your screen, the more power it (the monitor) will draw from the wall. If you don't see a power rating in watts, then multiply (the lowest) voltage by current. The monitor I'm using says 100 - 240 V, 1.5 A, max. That is, 150 W max. If I run it for 1 hour, I've used 150 Wh. If I run it for 7 hours, I've used 1050 Wh, or about 1 kWh. I pay $0.07 per kWh, so I pay $0.07 every 7 hours to use this monitor, or $0.01 every hour.

No, definately no. It makes no difference at all because the hdtv gets its power from its own power cable not from the cpu power. The cpu only transfer video signals with small amount of power to the hdtv. So, there's nothing you should worry about.