Unable to update, even after restarting.
Whenever you see this message you usually close the game client, wait a bit until the update finishes downloading then restart the game.
But sometimes, the updating process will never start, and you will be greeted with this dialog all the time with no way of getting rid of it, lest you reinstall the game —which is not practical when the whole game is around ~25GB.
So, what to do?
Note: The instructions presented here are for Manjaro Linux.
Close the game and the Steam client, then:
rm .local/share/Steam/steamapps/common/dota\ 2\ beta/game/dota/steam.inf
Restart your Steam client and it should start downloading that pesky update. 😄