The tv can get spoiled if the voltage given by the inverter is above 240v,all flat panel tvs in india are designed to operate at 240V max.
Also if you do not have a stabilizer and if your main line voltage is more then 240v again it can have a toll on the tvs life.
Your inverter should have a PC mode where the switching time should be faster,this should solve the problem of tv switching off,if not you need to get a computer UPS.