Do ATMs at banks in Mexico generally require cards with chips to withdraw cash?
You need to ask your bank if they allow non-EMV transactions, as it is the bank that will refuse the transaction and not the ATM.
All ATMs since the 70s/80s are equipped with the mag stripe readers, and most ATMs have been upgraded to accept mag stripe and EMV cards. So its not the ATM that will reject your card, it is your bank.
You run the risk of automatic rejection if you have a non-EMV card and are using it at a kiosk or vending machine, as these are high risk devices and thus have more stringent controls.
I have used both my mag-stripe only card and my chip card this year in Peru, Mexico, Spain, England, Wales, Ireland, and USA. Last year in Canada, Mexico, and Iceland. 2014 in Netherlands and Turkey. Someday, they'll stop accepting mag-stripe, but not yet. You might ask your bank why they are taking so long to make the switch.