Our marketing and communications department has our own team set-up, but is part of a larger organization (university). We want to be able to share a particular project board with a member of our university who does not currently use Asana, and who we don’t want to join our team.
Is there a way to add someone from within our organization as a guest on one of our project boards without purchasing a seat for them on our team?
Guests are users that do not share the same domain extension as you. Anyone with the same domain will take up a seat in your organization.
Adding to what @Christine_Bolton mentioned, you can learn more about the roles in your Organization in this article: Setting up an organization in Asana | Product guide • Asana 
I have the same need (also leading a department in a large university). So do I understand this correctly, the answer is “no, you cannot enable someone outside your team to view the project unless you buy them a seat.”
I don’t have the option to make our team space an Organization because I don’t (and don’t want to) control Asana for the full university. I do have times when I need an executive or a colleague to be able to take a look at the plan, to see how things are going.
Is there really no option for this?
Thanks,
Jim