The profitably of colonies varied wildly throughout time, some made a great deal of money, others did not. Some changed over time, some made lots of money, but it was pocketed by private individuals and did not make it to the state.

Generally the older a colony, the more likely it made money. By the 19th century nations were grabbing at whatever was left.

Germany's colonies famously contributed very little. Only Togo and German Samoa (modern Samoa) actually made enough money to pay for themselves.

Contrast this to to the early sugar colonies where the profits were so ludicrously high that thousands of Europeans were willing to risk the shocking death rate from tropical diseases. In early 18th century Jamaica, 200 per thousand would die a year, compared to 25/30 per thousand in England and 15/20 per thousand in New England.

Specifically dealing with India, it certainly made a great deal of money when it was the East India Company ruling parts of India, but once the company conquered most of the subcontinent, I believe profits dropped and the Raj didn't make a great deal either.

Still I'm fairly sure India was never a financial drain to the British Empire, but I don't have a particular source back that up.

It wasn't a prestige project though like the German Empire. The Company started out creating an army to guard warehouses, and as the Mughal state crumbled it found itself as the best organised and supported army in the subcontinent.

The conquered to secure trade, and to prevent the French conquering, and it all snowballed from their until they ended up taking control of the who subcontinent.