How long does drool rash associated with teething last?

It depends. Any rash will last as long as the cause is present. Some good ways to deal with teething pain are teething rings or frozen wash clothes. Use tylenol (acetaminophen) for severe teething discomfort only. Try applying neosporin to the rash and keep the area clean wiping away saliva frequently.
When tooth erupts. As teeth erupt babies can be fussy, sleepless and irritable, lose their appetite or drool more than usual. Try rubbing their gums with a clean finger, a small cool spoon or a moist guaze pad. A clean teething ring or pacifier will help.