1 and 2 are correct. 'look that' as used in 3 is not a valid usage, 'as if' would work instead.
3 is only a partial sentence; you could perhaps complete it as something like "He looks as if he has got so stressed that he might resign".
There are a few options for using 'look that', but they are unrelated to the original use you had in mind, and are quite unusual. For example
i) 'that' as a synonym for 'so', common in Northern English speech: instead of "He looks so stressed he might resign" you could say "He looks that stressed he might resign".
ii) as an imperative, meaning 'ensure that': "Look that you don't get too stressed".