I suspect you're hearing something relating to the electronics (possibly a miniature transformer) which drives the fluorescent backlight, or a sound from the backlight itself. You could check that by finding one of the new LCD TVs which is lit using LEDs instead to see if that makes a difference.
The highest pitch that you can hear varies with individuals and generally drops off with age. There are on-line web-sites where you can test your cut-off point, for example:
http://www.noiseaddicts.com/2009/03/can-you-hear-this-hearing-test/"
I'm 54 and the highest one I can hear now is the 12KHz one on that test. My wife can just hear the 12KHz too. My 9-year-old son can hear 17KHz and my 13-year-old daughter can hear 16KHz.
When I was younger, many TVs had an irritating whistle, which I found was at the horizontal line frequency (here in the UK), 625*25 = 15625 Hz. (At the time, I could also hear the whistle as an identifiable musical note which seemed to match that, which a quick calculation now says should be around a "B" about two octaves above the top note of a piano).
I've had problems in the past with CRT monitors issuing really irritating whistles, almost a scream, which is especially problematic when it's someone else's monitor in a shared office and they can't hear it. (Fortunately in that case the young lad called out to check it could easily hear it, and replaced it with one that I couldn't hear at all).
Most of the rooms in our house now either have CFLs or electronic dimmer switches (for bedrooms). The CFLs often seem to go through a nasty whistling phase while warming up, and the bulbs with dimmer switches emit an irritating hum all the time, although it's minimized when they are at full intensity.
My main hobby is classical music, and I've taken care all my life to avoid excessively loud sounds as I don't like being deafened, even temporarily. My hearing remains very good, both in terms of being able to hear faint noise and to be able to identify and locate it, but it does mean I tend to be very irritated by background noises such as disk drives and fans in PVRs and computers and whistling from my CFL bedside light. I also find I'm irritated by hum from transformers for electronic equipment even when the equipment itself is turned off.
. Egads it's annoying.
That was when I first realized that I had tinnitus.