They do not ask for any card details, I've used the vouchers 2 times. They adjusted the voucher against the bill and the remaining amount I've paid using HDFC TATA NEU card. The reservation front office do not care from where we got the vouchers, it should be Stay/Experiences vouchers that's it.
You can still call the Taj Central Reservation office and ask them if the card is required when redeeming the voucher or not.