Will posts that are disapproved still be able to be published?

Not if you don't want them to! In Lately, you can now choose to only allow approved posts to publish through a toggle on your Team page (Planning -->Team). 

Here, there is a new Team Setting section below your Manage Team Members library. If the toggle is switched on, scheduled posts in the dashboard won't be published unless they are approved. Also, posts that aren't approved won't be able to be published manually. 

To note, only the owner of the dashboard can control that toggle. So the owner of the dashboard will have complete control over their Team Settings and the management of their approved/disapproved posts.