Just to be clear when I have referred to Quasi Sine Wave Inverters, it includes "Modified Sine Wave Inverters - they are the same thing i.e not Pure Sine Wave.
Yes QSWI is more efficient than a Pure Sine Wave inverter at converting the power which will mean it will drain a battery less quickly. But there can be other effects, which means the appliance you connect to the QSWI may not work as efficiently.
I have already mentioned transformers and motors getting hotter compared to pure mains, Some can also sometimes make extraneous noises as the stepped changes in voltage excite electromechanical resonances or phase reversals in windings. In the worst instances motors may run slower, I have witnessed a record player [- do you remember them?
] synchronous motor that ran slowly and mechanically buzzed and got hot when it was run on a QSWI.
There is no simple way of predicting which appliances might react badly to a QSWI, and it also could depend on the make/model of inverter.
With the advent of low cost and wider range of available high voltage components there has been a rash of cheap QSWI's mainly from the far East. They often exaggerate their output capabilities and produce poor output voltage regulation and waveform smoothing. I strongly suggest if you are looking at such a device you would do well to go to reputable supplier and avoid the cheap imports which could damage your probably expensive TV.