If I worked at a local place (CT) but the W2 lists an AZ address, do I have to file AZ State taxes?

If you lived and worked in CT, then you only pay CT tax.   Many employers have corporate offices in other states from which they mail the W-2.   You do not have to pay AZ tax just because they mailed your W-2 from AZ.

If the AZ address is your employer's address, no, you don't have to file an AZ state income tax return if you are not a resident  of AZ,  

• Remote workers do not have to file nonresident state tax returns unless they physically travel to another state and perform work while they are there. 
• Generally, you’ll need to file a nonresident state return if you made money from sources in a state you don’t live in.

If you are not a resident of AZ and don't physically work in AZ but your employer withheld AZ state income taxes, you'll have to file a nonresident income tax return to get a refund of the AZ income taxes that were withheld from your pay.

• Please see How do I file a nonresident state return?