Cards like Dhol Chants state that if you pass the test, you discard the card and receive another one. It doesn't say what to do if you fail the test. Do I discard it also, or do I leave it in that room for the next action turn/next investigator?
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.