I am a dual citizen with CA and the US. I currently reside in CA and have done so for the last 15 yrs (I am 22). I travel on my CA passport. This fall I will be traveling in the states for 89 days for what is considered a "non-academic institution". Do I need a student visa? I have not renewed my American passport in years.
If you are a US citizen, regardless of what other citizenships you might have, you can enter the US and do whatever you want, for as long as you want, without any kind of visa. Just renew your US passport and use it to enter the US.
(If you've previously traveled to the US on your Canadian passport, technically that was illegal. It doesn't have any practical consequences, as far as anyone seems to know, but the US does want you to use your US passport.)
You do not need a visa. Furthermore, you cannot get a visa, except perhaps in a situation where your US citizenship may be in doubt and you need to travel urgently. Because you have an expired US passport, that exception does not apply to you.
From the Foreign Affairs Manual of the US State Department, emphasis added:
9 FAM 202.1-2 VISA-RELATED ISSUES WITH U.S. CITIZENS
a. Processing Visa Applications for Aliens Who May Have a Claim to U.S. Citizenship:
(1) Nonimmigrant Visa Applicants: You may not issue a visa to an individual who has been determined to be a U.S. citizen. However, if a nonimmigrant visa applicant with a possible claim to U.S. citizenship is unable or unwilling to delay travel until he or she has been able to obtain documents to establish that status, as determined by the posts citizenship and passport officer, you may presume that the applicant is an alien pursuing a nonimmigrant visa application. If you find the presumed alien eligible to receive the visa then you may issue the visa.
As a general rule everywhere in the world:
- States are mainly interested in your citizenship in the sense of whether you are their citizen or not.
- Their main stand about your other citizenship is "none of our business".
- If you are a citizen of a state, you can enter or leave it any time.
There are many, many, many exceptions, but in practice you can face them rarely.
For the U.S., what matters is whether you are a U.S. citizen or not. If you are not, then it could matter whether you have some other citizenship which makes your case easier. But being a U.S. citizen, it doesn't matter what other citizenships you have.