Do I need proof of travel to enter India with a one-year multiple entry tourist visa as a Lithuanian (EU citizen)?

You seem to have missed an essential element – the visa states (third line of the bottom of your photo):
Continuous stay during each visit not exceeding 90 days – you have to leave before the 90 days are full. You can come back several times during the year, but you can not stay all year.

With that, having a booked return trip makes a lot more sense. However, as others have said, as a rule they do not ask for proof of booking.