I have a role with access to several CPTs. They can add and edit posts but cannot edit or select information in the built in or custom taxonomy metaboxes.
The odd thing is, they can add/edit the taxonomies from the admin (left side) menu, just not on the edit-post.php page.
Nevermind I figured this out. But now when I search in the admin area say for a particular post to edit, instead of results I get kicked back to the dashboard.
Redirection to dashboard issue: I suppose that you limit a role access to some admin menu items via “Admin menu blocking” add-on with “Block no selected” option. If ‘Yes’ read this section of the documentation page.