So my US passport expired a few weeks back and I expedited the renewal process, however, I'm not sure it will arrive in time for me to fly out the Mexico in a couple weeks. I have dual citizenship for the US and Ireland, and I'm wondering if it would be any issue for me to leave the US with my Irish passport? I know I need my US passport for reentering the US, so I can either have my sister fly it down when she comes to meet our family as she'll be leaving a few days after me (or I believe I can have it FedEx-ed to a secure place if not)... Aside from the obvious question as to whether I'll get my US passport back in time for my departure, can anyone please tell me or confirm that exiting the US without my US passport won't be an issue?
The US doesn't have exit controls, so you won't have a problem usually. The regulation doesn't have a penalty to be enforced anyway. You don't need to be worried about the US here, but whether the airline will let you board or not, which depends on if you'll be able to enter your destination. I assume that on an Irish passport, you'll be able to enter visa-free in Mexico, although under a bit harsher conditions than with a US passport. You would probably get a migration/tourist card, which you have to keep on your person and give it back when leaving.
can anyone please tell me or confirm that exiting the US without my US passport won't be an issue?
It's not an issue. US has no exit controls. You can enter a third country with any passport you like.
I know I need my US passport for reentering the US,
Since it's so hard to renew passports, there is a Covid exemption in place: https://www.state.gov/return-travel-on-recently-expired-u-s-passports/ You probably board a US bound plane with an expired passport, although I would check with the airline first.
Quoting from the U.S. DEPARTMENT of STATE - BUREAU of CONSULAR AFFAIRS:
U.S. nationals, including dual nationals, must use a U.S. passport to enter and leave the United States.
The US doesn't have consistent exit controls, but some users online report of random checks being carried out by Customs. Personally I've never seen any, or know anyone who did, so I'd assume they're fairly rare.
Assuming you find a workaround, or leave anyway, I would strongly suggest bringing the expired passport with you.
DO NOT HAVE YOUR SISTER CARRY IT FOR YOU. Generally speaking, you're not allowed to carry someone else's password, if they're not traveling with you. I know in the US to legally carry it she would need an authorization letter from you, but I'm not sure it would be enough to cross the border. It might be illegal.