Emacs, org-mode, evil-mode - TAB key not working

Try

(setq evil-want-C-i-jump nil)

in your ~/.emacs before

(require 'evil) 

Evil has, in evil-maps.el

(when evil-want-C-i-jump
  (define-key evil-motion-state-map (kbd "C-i") 'evil-jump-forward))

That should give you org-mode Tab functionality back

I have almost no experience with terminals. However, I know that TAB is equivalent to C-i. Maybe that one would go through the terminal? If that works, you could add some key bindings for every TAB operation?

Try maybe C-h k TAB as well to see if TAB if sent on the wire.