I am having an issue on a VisualForce page where my queue is not auto-updating with an ownerId based on the profile I'm on. It's not throwing an error anywhere and giving them the 'View All Lookup Record Name' permission fixes it. However, I do not want to give that permission to the profile so is there any way to give the profile enough access so they don't need this permission. This only happens with queues as ownerIds, not users and I already tried giving them all access to the Group object.
If your VisualForce page is using a standard controller, then the user's permissions are being used. The page will assign the case to the queue only if the current user has access to said queue.
You can use a custom controller that is explicitly not taking sharing into consideration, or you can set up a custom process (flow/process builder) that runs when the case is created by a user with the specific profile you have. The process can automatically assign the case to the queue if you want.
Assuming your users should not have access to the queue, I think the best approach is the automated process one, without code.