Would anyone be able to offer advice?
I'll be travelling to Austria for just barely over three months, on my American passport.
As an American, I'd need a visa. I'm also a EU citizen though, and wouldn't need one in that case - unlimited stay.
My EU passport has expired, so I can't use that one to enter the country.
Would I run into trouble at border patrol when I leave, if they saw that I was there for over three months? Even if I can prove I'm an EU citizen?
If you enter the Schengen area of which Austria is a member on your US passport, you'll get a 90 days stay without a visa. If you leave on your expired EU passport, the US passport won't be scanned or stamped, and it will look as if you never left. This could result in a lot of problems the next time you plan to enter Schengen.
Best is to get your EU passport renewed or to try and enter on your expired EU passport - it might just be possible - or to stick to the rules and leave after 90 days on your US passport.
Entering and leaving on different passport is likely to lead to trouble.