Here's a (currently very short) list of bank ATMs that serve Euros. More will be added later.
Raiffeisen Bank - Central Bus Station, Rruga Egnatia/Grigor Durrsaku, Rruga Alexsander Goga
Handily, the main branch of this bank is right at the end of the bus station on the corner of Rruga Adria and Rruga Egnatia. Fee to withdraw: €6
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. Fee to withdraw: €7
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Opening times:
Times to be confirmed. It's to be expected that ATMs work 24/7, however, those banks with ATMs in internal foyers may refuse access overnight.