Inverters today are quite efficient, in the 85 to 90% range. Televisions have some kind of power supply built in to suit their electrical needs. They do not inherently operate on 12 volts DC, so even those 12 volt
units may convert their power system from 12 volts.
Bottom line, there could be some efficiency gain on some Television Sets, but there is no guarantee.
That said, the WFCO sine wave inverter supplied with the Airstream inverter package requires almost 2 amps just to be on, so it is not one of those modern efficient inverters I mentioned in the first paragraph. Many of the small, less than 200 watt modified sine wave inverters you can buy for under $30 are a better bet, as they have a standby loss of under 0.4 amps. They will also operate your television just fine.
To all: don't leave your inverter on when not using it, and don't run it to do things like charging your phone. The standby losses of most pure sine wave inverters are quite substantial. A 2 amp load 24 hours a day is 48 amp hours. One group #24 battery only has a total capacity of about 75 amp hours.
I know that my statement seems contradictory. An inverter running a television is ok, but only if you shut it off when not in use, and it would be best if it were not a large pure sine wave inverter, as most of them have substantial running loss at low loads. The 300 watt Morningstar Pure Sine inverter is one of the few exceptions I have found to that statement. Small modified sine wave inverters are actually better for low load use.
Ask a question, get too much information, I know. Sorry, I just have to be complete.