What happens to the IP when the company is shut down?

Contingencies such as you have outlined are normally addressed when the company is originally formed. IP is property, it is assets, and therefore obviously has value.
If, for example, the company was a corporation, such contingencies probably would have been addressed in the Articles of Incorporation. It would seem to me that somewhere along the line, the property, and its ownership and rights to it, would have been addressed.

That is the million, or in some cases billion, dollar question. As Steven stated, what you describe which we call "intellectual property" is property, i.e. assets, that belong to the business and need to be disposed of or allocated as would any other property acquired by that business during the course of its operation or start-up. Ideally, as Steven, the disposal of this property was addressed in an agreement or Articles of Incorporation when the company was formed. However, as lawyers who often get to these situations only when the fire is already burning and needs to be put out, such an agreement or bylaws typically tend to be nonexistent and what you describe about one or more founders/partners/shareholders taking the property or know-how and working on it on their own is common. And once one or more of those founders make money with the IP to the exclusion of the other(s), that's when the other often gets upset, sues, and a huge mess ensues. So hopefully, if that agreement or arrangement is not yet worked out, it can promptly be worked out ASAP before the parties find themselves in litigation. A good corporate lawyer in the relevant jurisdiction should be consulted ASAP.

In order to protect your IP, and assure the value of it, independent of the sake of the company, you could consider an escrow-arrangement with the company-stakeholders (shareholders, investors, team-members, customers) as beneficiaries.
The developing company deposits its knowledge and technological assets with an escrow agent who acts as a trusted third party.
In case the company is shut down, or at the occasion of other predefined incidents, the IP remains available to these beneficiaries. They can continue using it and thus benefit from it.
From the point of view of the user of the technology (e.g. customers), this is a sound business continuity measure, part of any serious corporate governance policy.