I was born and raised in California but I have dual citizenship with Ireland. My Ireland passport has not been stamped, ever. I don't have an American passport due to arrears on alimony. If I leave from California to go to Thailand and only have my Irish passport and California driver's license, will I encounter problems getting back into the United States some two weeks later?
You are legally required to enter AND exit the USA on your US passport as a citizen of the USA.
When you check in for your flight to Thailand the airline is required to verify that you have proper documents for travel. Your Irish passport has not been used to enter the USA, so they can not record you as exiting on it. And if they realize you are a US citizen they can not board you without your US passport. So basically it will be nigh impossible to board your flight out of the USA.
Assuming you do manage to talk your way onto the plane, likely your Irish passport will be flagged for having exited the USA without having entered, so CBP may have a "few" extra questions for you upon your return.
And since you are prohibited from having a passport due to alimony issues, chances are that had a don't leave the country order exists, which might land you in court upon your return.
Bottom line, traveling to Thailand at the moment is not a good idea.