Depends what you mean...
Are you talking about getting beaten in duel because avatar prevents you from using spells that are crucial to you? If so, you could attempt to avoid the enemy until avatar ends.
Are you asking if there is a counter skill or build for facing an enemy using Avatar? If so, there is no definitive answer, because CHA is very unlike Rock-Paper-Scissors; the appropriate counter strategy would depend on other aspects of the opponent. It wouldn't be too useful if I suggested something like "well, avatar makes them spell-immune, so just do a build that uses no spells." As you may already know, there are effects that seem spell-like that will still affect an enemy under the effects of avatar, such as slow ward, poison, and certain effects of ultimate spells, but I wouldn't call any one of these a "counter."
There may be no answer to your question, at least no answer of the kind you're looking for... not everything has a clear or simple "counter," such as a single skill or item. Rather, you ought to look to think of countering someone in terms of many different adaptations to your style during the game, which all act to mitigate the effectiveness of the opponent.
If you're looking for general tips against any enemy using Avatar, a few things do come to mind, though you should always be sure not to considering just one aspect (like a certain skill) of the opponent when designing your strategy...
-Be mindful of skill cooldowns. Your opponent is probably going to use Avatar during each duel; this will make it easy for you to determine when that skill is cooling down. Use this to your advantage (ie. realize that if you can use spells to gank your opponent, they will not be able to escape if you make your attack when Avatar is cooling down)
-Be mindful of the advantages that Avatar provides, and prepare to mitigate these if possible. (ie. Avatar allows you to bypass the effects of spells that bosses use, which makes it possible to take down bosses earlier; therefore, be ready to hinder your opponent should they attempt to do a boss run early)
These, however, are rather vague suggestions and are really nothing more than general playing tips... a more specific answer would require more parameters.
