The Drawn Together drinking game is mentioned in "Charlotte's Web of Lies", the original description being "Drink every time something gay happens." The producers explain more in the commentary on the episodes in season 3, specifically "Lost In Parking Space", where they persuade people to play the "Drawn Together Not-Drinking Game" (Obviously phrased this way to avoid liability.)
On the Party In Your Box version of the release, you can select the "Get the DT's Drinking Game" for certain episodes. (Still avoiding liability, they claim that "The only thing funner than watching Drawn Together, is watching Drawn Together while drinking milk.") Instructions are included for the episodes "Requiem for a Reality Show", "Clum Babies" (the game is for the DVD commentary rather than the episode itself), "Captain Hero's Marriage Pact", and "Wooldoor Sockbat's Giggle-Wiggle Funny Tickle Non-Traditional Progressive Multicultural Roundtable!". "Captain Hero's Marriage Pact" also includes a "Sad Loser" version, for people watching alone. It consists of a baby bottle showing up on the screen when it is time to drink, often accompanied by an insult of some sort.