Actually, it appears ‘BuddyPress Reorder Tabs’ is overriding it. I’ve got a new question, but I’m going to start it on a new post. Feel free to delete this.
I’d like to ‘unresolve’ this topic, if possible–It turns out it’s NOT that plugin that is overriding the reorder tabs facility. Any other reason why it wouldn’t be saving the reordered tabs?
I’ve just tested the latest WordPress, BuddyPress and BPGE versions – saving tabs after reordering does work (no 3rd party tabs reordering plugins, no related code in a theme or bp-custom.php, on 2015 default theme).
Have you saved changes after reordering? It’s stated in the description:
You can reorder here all navigation links in this group. The first item will become a landing page for this group. Save changes after reordering.
Note the last sentence.