Are there free or donation-based ashram stays in India?

Yes, a few. Sri Ramanasramam (Tiruvannamalai) provides rooms and meals free of charge and welcomes donations. Rooms are reserved for devotees of Sri Ramana, you must apply online, and only about 25% of requests can be accepted. Vipassana courses in S.N. Goenka's tradition (for example Dhamma Giri) run only on donations. Donations are accepted only from people who have completed at least one 10-day course. The Divine Life Society says visitors can stay and eat at its Rishikesh headquarters without charge. The Society also asks visitors to write for and receive written permission before coming, because space is limited. Paid options also exist: Sivananda ashrams charge daily rates that include classes, food and accommodation.