Leaving the US on a different passport to visa

You implied in a comment that your main worry is being recorded as overstayed. For the purpose of tracking overstays, the airline sends your passport number and other information to CBP electronically. Your exit date is then recorded in your admission record, which is the data used to develop potential overstays.

Usually this is something that can be cleared up the next time you travel to the US or apply for a visa. The system isn’t perfect, and some exits (even with the same passport) are not always recorded, so an overstay is not automatically assumed. But of course you don’t want to be in the awkward and uncomfortable situation of having to prove to CBP that you did not overstay.

There are two things you can do to reduce the risk to yourself:

1. Alert the airline to the situation when you exit. They are the ones who send exit data to CBP. Perhaps explain that you would like them to use your passport with the US visa for exit-tracking purposes, but your other passport is the one that allows you to travel to your destination. Depending on their IT systems and how busy and knowledgeable the staff that day is, they may be able to understand the situation and enter your information correctly. Have your I-94 printed out and ready when boarding. (In the “old” days before electronic I-94, the airline would collect paper I-94 forms for forwarding to CBP, and they still have to do this occasionally for the few paper I-94s that are left. The airline staff may be able to take your I-94 printout and submit that to CBP.)
2. A few days after you have departed the US, view your travel history using CBP’s online system. Try both passports. If your exit was correctly associated with the passport your visa is in (and which you entered the US on), you are very unlikely to have problems. If it wasn’t, you should prepare evidence that you left the US.

You wrote in a comment to a now- deleted answer:

My visa expires in a month, and the purpose of the trip is to renew the visa at the US consulate in my home country, Ireland. My visa is tied to my Australian passport. I just want to make sure that if I leave the US on my Irish passport, that it will not be considered "overstaying" my US visa.

Since your Australian passport allows you to enter Ireland without a visa, I would suggest that you show the Australian passport to the airline at check-in, ensuring that your departure from the US is matched to your most recent entry. Then when you arrive in Ireland, show your Irish passport to enter the country.

As an aside, you may be interested to know, if you don't already, that you are not required to leave the US before the expiration date on your visa. The critical date is the one on your I-94, which is usually also written on your most recent admission stamp. If instead of showing a date it reads D/S then you can stay as long as you are eligible for your current immigration status.