After failing the test on Dhol Chants, does the card stay in the room or get discarded?
If you fail the Test on a card, the investigator holding it still keeps the card UNLESS SPECIFICALLY TOLD TO DISCARD IT. 

Your penalty with the tomes is that you've blown your action. The tome remains in your possession, so you can try again.

Reading the question, it seems like there's confusion. When you discover the tome by exploring, it's just equipment you pick up. You have to use another action (normally on another turn) to actually read it, but you can carry it around with you till you find a safe place to do so.