It definitely depends on the location. Lots of girls wear tank tops and sleeveless shirts in places like Delhi, and it's fine. In rural areas, it's not such a good idea. It's also, surprisingly, often not such a good idea in places like Varanasi, which seem like big cities, but in many ways are just huge, sprawling villages, with lots of gossip.
Even where it seems fine, there are still drawbacks to wearing tank tops and sleeveless shirts. Most Indian women wouldn't do it, so lots of people take it as a mark of respect when tourists don't do it, either. The other thing is that wearing long, loose sleeves can actually be better in the sun than tank tops or sleeveless shirts, as it keeps the sun off you, and can sometimes be cooler. For instance, the people who live in the Thar desert wear long, loose, flowing clothes, just like the Bedouin tribes.