@fBoyle-- No! Don't do that, that's a bad idea!
The alcohol in regular mouthwash will irritate the piercing and possibly make the infection worse. Plus, it will burn a lot!
You can however use an alcohol-free antiseptic mouthwash to clean the infected part inside your mouth. Just as the article mentioned, saline soaks are also great. You can get that stuff at any pharmacy and it's sterile, so it's best for cleaning infections. Aside from that, you need to wash your hands often if you touch your piercing with your hands.
If your piercing keeps oozing greenish colored pus, is swollen, red, very painful and you start having other symptoms like fever, swollen lymph nodes or fatigue, it's very serious. You definitely have to see a doctor and take antibiotics in that case like @alisha mentioned.
Hopefully, the saline soak and mouthwash will do it for you. Good luck!