Here's a (currently very short) list of banks that serve Euros. More will be added later.

Raffeisen Bank - Central Bus Station

Handily, this bank is right at the end of the bus station on the corner of Rruga Adria and Rruga Egnatia.

Raffeisen Bank Location

Intesa San Paolo Banka - Sheshi Liria

Right across from the Great Mosque, although I've found the ATM to be out of action a couple of times.

OTP Bank Location