Yes, there are drinking games involving Saké, including at least one with a very long history. Care to partake of such a game?
Kyokusui-no-en, also known as gokusui-no-en and which translates as "party held by a meandering stream," is a game where the participants must compose poetry within a certain time limit. The game is held outdoors, beside a stream or slow flowing waterway.
In ancient times, March 3 (using the lunar calendar) was a time when the Chinese would purify themselves, washing away their sins. Over time, this developed into the game of Kyokusui-no-en and became very popular in the fourth century. The Japanese later adopted the game, which became popular with the nobility during their Heian period (794-1185).
The participants, aristrocrats and artists, would sit by a stream while lacquered Saké cups would be sent floating down the stream. A person had to write a tanka, a short poem, on a piece of paper before the cup passed him. If he failed to do so, then he had to drink Saké as a punishment. (Though I would not consider that a real punishment!)
In the present day, a Kyokusui-no-en Festival is held in certain areas of Japan on the first Sunday of March. They still play this poetry writing game, though sometimes with slight modifications.
You may not have a stream by your home, but you could try something similar. Just use some type of timer, giving each person a set amount of time to write a small poem, maybe a haiku. If they run out of time, they must drink some Saké.
Have you played any drinking games with Saké?