Does the Great Old One warlock's telepathy from the Awakened Mind feature take an action?
It doesn't seem too clear to our group and we have basically ruled it as equivalent to talking.
Awakened Mind just allows you to telepathically communicate. You are able to communicate whenever you like as part of "Other Activity On Your Turn" PHB 190:
Your turn can include a variety of flourishes that require neither your action nor your move.
You can communicate however you are able, through brief utterances and gestures, as you take your turn.
Be careful of using it too much though, because in the last paragraph,
The DM might require you to use an action for any of these activities when it needs special care or when it presents an unusual obstacle.