From Homestar Runner Wiki
Listerine is a brand of mouthwash. Coach Z can occasionally been seen using it to get drunk. He was first seen drinking it in The Best Decemberween Ever, where he offered it to Homestar Runner and it caused him to pass out at the end of the toon.
In the email senior prom, because Pom Pom was at a cooler private school prom, Homestar accepted the Prom King award on his behalf while gargling some Listerine in order to sound like him. At the same time, Coach Z was spiking the punch with it, much to Strong Sad's disappointment.
While it is possible to get drunk from heavy consumption of Listerine, the denatured state of the alcohol in it would probably cause the drinker to regurgitate before it could even take effect.
In A Decemberween Mackerel, Homestar, Marzipan and Senor Cardgage sing a Christmas carol with the line "Coach Z's been drinking non-specific mouthwash-rine". An arm, brandishing a bulky legal contract, appears from off screen, menacing Homestar to sign. This suggests that either the makers of Listerine may have pursued The Brothers Chaps over trademark infringement, or that TBC are pre-emptively avoiding such action.