Is there anything stopping the Overlord from discarding each Ovelord card he picks up to gain lots of threat tokens and making the players lose 3 conquest tokens instead of playing any? Is this a common tactic? What are the disadvantages to doing this?

Discarding cards won't allow you to cycle your deck any faster. You can only discard a card for threat when it's in your hand, and doing so doesn't let you draw a replacement card, it just moves a card from your hand to the discard pile and gets you threat tokens.

You can hypothetically discard every card you draw for threat, but all that would do is get you a bunch of threat, which is worthless unless you have something to spend it on. The discard mechanic is basically just a way of letting you choose which cards are most important - you discard the less desirable cards to pay for playing the more desirable ones.

Thank you that sovles it for me.