For example, when a list is updated, a user can subscribe to the list so they get emails about the changes to the list. I can remove a user from getting those alerts, but the user can simply go back to resubscribe again. I want to stop that user from subscribing to certain lists. Is there a way?
Create a custom SharePoint Group (call it Restricted Alerts or whatever). Put your user or users into that group.
Create a new permission level where under List Permissions the "Create Alerts" permission is unchecked. This is done by going into Site Permissions | Permission Levels. You can copy an existing permission set and just make that change.
Now give that SharePoint Group that Permission level on those lists.
That should remove their ability to create alerts assuming they are not given some additional permission level via another group or directly in those lists.