Listerine and other mouthwashes contain around 26.9 percent of alcohol which if you drank enough of the Listerine or other mouthwash it could make you drunk.
But if you drank enough mouthwash to make you drunk you could die from it as Listerine and other mouthwashes contain other chemicals besides the alcohol which is supposed to make the mouthwash undrinkable and those chemicals could be fatal if ingested.