After completing a B.Tech from NIT Trichy, is it advisable to pursue an M.Tech in IITs or IISc?

No, it's certainly not a wise decision to pursue an MTech after completing BTech from NIT Trichy. The courses and the approach are similar, giving you no newer perspectives and wasting 2 years as well as the money spent. A better alternative is to go for an MS or Ph.D. abroad instead. That will be the true utilization of your money, knowledge, and time. 

However, if you are hesitant about going abroad, an MBA from an Indian B school is also a great option.

An MTech in IIT should only be considered if you have a low CGPA and have no project achievement up your sleeve. The MTech course will give you a second chance to improve your grades and work well on projects improving your profile as a whole. This will help you open better Ph.D. options. But if you are not planning to do one, then MTech is again of no use.