I gotta admit, I strongly disagree with the advice that boils down to "nerf this ability". Whether it be things like "kill it frequently" or whatever. IMO, you will get much better results by compromising what you think is fun with what the player has indicated will be fun.
Now, there is the practical issue of time. If you're spending half the session with the warlock player doing a one man show basically investigating the entire adventure, then sure, that's no fun for anyone. D&D isn't a spectator sport. But, the solution here isn't necessarily to nerf the warlock, but, rather to adapt and overcome.
You know that the Warlock player is going to do this, so, prepare for it. Write up a handout beforehand for what the quasit learns. Ok, here's a map of where the quasit went (show an edited map of your adventure, leaving off stuff that the quasit couldn't have learned - traps, secret doors, etc. Heck, use that map as your handout. Freehand the map - it doesn't have to be perfectly accurate, it's not like the quasit is pacing off rooms. Label the map with what the quasit saw inside - this is a guard room with 4 orcs, this is common room, this is a garbage room, etc. Did the quasit overhear stuff? Well, does the caster speak that language? It's not like languages are that common - lots of critters speak stuff that the caster doesn't, in all likelihood. So, maybe the warlock learns half a dozen things - PLOT HOOKS FTW!
Now you have a cool little handout to pass off to the warlock player to share with the group. Cool, you just laser beam focused the entire group's attention. Hey, maybe the orcs are talking about something in those rough caverns back there that scares them, but, they haven't seen it, and the quasit didn't either (ochre jelly, invisible stalker, mimics, pretty much any "ambush" monster).
Really, this is no different than when the mage gets Arcane Eye. Arcane Eye is every bit as powerful as this quasit combo. Great. Don't fight it. USE it. Instead of your players wading in blindly and just reacting, now they get to be pro-active. I think you'll find that once the players make that transition, your game will be very fun.