Hi there!

I'm having a problem with my high e string on my Yamaha acoustic.
Whenever I play a chord that involves me pressing on the third thread (or the fourth with capo on one, fifth with capo on two etc.) the high e string is flat. The string itsself is in tune and it's fine with every other fret. I thought at first it was my capo that was the problem because I noticed it while playing G chords and Dsus4 chords, whenever my capo was on, but it's happening without my capo too.

I haven't had my guitar re-strung in a while, could this be the problem? The other strings are fine.

Occasionally a bad string will cause problems with intonation because it doesn't vibrate evenly or some such thing...
That would be your first step. New strings. Then, play each note up the fretboard to make sure you haven't got any high frets or other problems.
Finally, check the intonation... You should get the same note (an octave higher) at the 12 fret as the open string, and also the 12th fret harmonic. If it's high or low, you may have an intonation problem.