More than likely, yes. That's a pretty heavy load for what you described... and a little dangerous. Double check your inverter to ensure it is capable of delivery the juice your TV needs. Not all 400w inverters are created equal.
I would really consider running a dedicated circuit for a higher demand electronic like that. Among other things, you will want to consider wire gauge and distance. 12v current isn't too fond of travelling
So all told, it could be the inverter itself isn't capable of providing the needed amperage, or, it could be that the wire gauge and distance are such that the inverter is not able to draw enough juice from the line to perform properly. In the later is where it can be dangerous. It will TRY to draw enough to meet demand which can lead to overheating and melting wires.
It's possible too that your 23" just doesn't like what it's getting and pure sine wave inverter might work better. Have you tried testing your inverter/TV set up from your vehicles power plug? Off the bat though I would be most concerned about where and how you tap into your 12v source.