In an episode of the Simpsons…

In an episode of the Simpsons that aired in 2003, Homer gave his email address as The episode’s writer, Matt Selman, signed up for the ChunkyLover53 email address beforehand and within minutes of the show’s airing found his inbox packed to its 999-message limit.
There was also an episode of Futurama where Bender gave out his email address ( if you wrote, you got an automated message back from bender:

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