Do the .NET app pools need to be running for SharePoint to function as long as all the sites run on their own separate app pools?
If you're talking about the out of the box app pools (.NET NNNNN, DefaultAppPool), then no, and I usually delete them. Of course if something else is running on your SharePoint server that does use them, then you will not be able to delete them nor should you.
In IIS Manager, if you go to the Application Pools node, you can see the number of applications assigned to each App Pool. If the value is 0, it is safe to delete.