How long can I keep my carafe on the warmer? I've heard that the heater plate of my coffee maker will give my coffee a "burnt" aftertaste. But about how long do I have after brewing? Is the effect more or less immediate, or will the effect not be noticeable for a few hours?

I'm using a Cuisinart DCC-2650 with an adjustable warmer.
It would depend on a number or factors including temperature of the warmer and humidity in the room.  However, personally, I can begin to taste a difference after as little as 45 minutes on the warmer.  I tend to drink two cups of coffee pretty quickly to avoid this.  

Coffee is a product always best fresh, but how long before your palate can detect that it isn't fresh any longer is a matter that varies by individual.  My wife will drink coffee from a pot that has been sitting for four hours plus; after about an hour and a half I'll throw the pot away and make another.  Your own mileage may vary.