You need to fix the plugin that is pulling in the data so that it will "strip slashes"

It might be worth trying to preceed the apostrophe with a \ (so it would look like don\'t )
It isn't likely that it will work but if fixing the theme/plugin isn't and option it is worth a shot.

What Kirk and Kevin said is right. Another thing you could try in the meantime is use a HTML ASCII character for an apostrophe, which is this:
'
I know this is not ideal.