You will need to clean any control/adjustment potentiometer in the tweeter feed or replace.It is acting like a diode.
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It is probably not inside the tweeter.
The comment was the FEED to the tweeter has a problem.
I agree that is the most likely problem.
You need to clean the connections inside the speaker, the wiring to the tweeter, the crossover connections to the tweeter, and most tweeters have a tuning pot to alter the volume of the tweeter as conpared the the rest of the speaker, and clean that control.
A good product is Caig labs Oxy ? clean? or similar from RadioShack. just clean the connectors and stick 'em back together. Should work.