You can include any Unicode character in a tweet, as the full character set is supported in Mathematica (see here.) Since Twitter doesn't support attachments, (except images, for which you include the url in the tweet) the best solution is probably to use the Wolfram Cloud:
CloudExport[ExampleData[{"Sound", "AltoSaxophone"}], "MP3",
"Permissions" -> "Public"]
The permissions option is required to make the file visible to all users.
Alternatively, depending on the type of file you want to display, you might be able to render it into an image.
