There's a Citibank office with an ATM in the financial district on Sathorn Road in Bangkok. I can't recall for certain whether I saw one on Level 3 landside at Suvarnabhumi, but I may have. I definitely do not recall seeing one in Phuket.

I imagine you're looking for a Citi ATM because you're a Citi account-holder. For the benefit of others reading this thread, all of the major commercial banks in Thailand--Siam Commercial Bank, Bangkok Bank, Kasikorn (Thai Farmers') Bank, Krung Thai Bank, Bank of Ayuddhya, and I think even Thai Military Bank--readily accept ATM cards using the major international networks such as Plus, Cirrus, etc. That said, most international banks have not penetrated the retail banking market in Thailand yet, although Singapore's UOB is building a presence.