Paladins are not Druids
The Oath of the Ancients has nothing to do with nature; it is about life and light.
A Green Knight has committed to fighting for a world where things can live and grow and flourish. Here, "things" includes plants and animals, true, but also includes people and ideas and art and concepts and societies.
Green Knights use growing things as their symbols because those things are the opposite of death and decay, which symbolise what the Knight is fighting against.
They do not necessarily have to kill wicked people and monsters. A Green Knight's focus is providing an environment where the light can prosper. If that can be done without killing (perhaps by moving the creatures on), then that is OK (though individual Paladins may differ on the level of restraint they feel is acceptable). If the evil creatures can be redeemed and brought over to the side of the light, that is (in my opinion) the best outcome. (Hat-tip Molot.)
Examples
Travel with evil aligned characters without any attempt at reforming them?
No. Green Knights are supposed to stand against wickedness, not stand beside it.
If there is a greater evil, the Knight might put up with a lesser evil, but only temporarily. Even so, the Knight should at all times try to redeem the wicked, through the example of their own actions (Be the Light).
Forego forgiveness in favor of killing someone who betrayed him?
Maybe - there should always be a hope of redemption. However, a strict Knight might feel that one transgression is one too many. Any Knight might refuse forgiveness to one who has betrayed them several times.
Overlook the evil and wrongdoings of man? (or any sentient race)
No. Again, the Knight is supposed to stand against wickedness, not stand aside.
They are not, however, stupid. They won't engage in pointless attacks. Throwing your life away for nothing violates the tenet of Preserve Your Own Light.
Inflict injury on someone for lying to him?
Maybe. A very strict Knight might be justified in punishing lying with injury.
Media Figures
I think Wonder Woman is an example of a Green Knight.
From an interview with Patty Jenkins, the director:
Jenkins goes on to describe Wonder Woman’s worldview as, “I believe in the betterment of you, and I believe in you, and I believe in truth. And I believe that you all can be better, and I will fight to protect you. But I also believe in better than that.”
Best Answer
There are evil fey. The green hag and sea hag are evil fey, while the night hag is considered a fiend. This is a change from earlier editions of the game, where hags were monstrous humanoids. In the Eberron setting, all hags are considered fiends. These enemies are the primary reason why a paladin would need such an ability.
The paladin may also occasionally come into conflict with non-evil fey, such as the neutral dryad and satyr. A good paladin is unlikely to come into need to use their power against good fey, but technically still has that power. Fey is a very small category of creatures, so the addition of fey to the paladin's ability is not a massive power boost.
There is also a thematic link between fey and fiends, both being traditionally weak to metals such as silver or cold iron. A hero equipped to defeat fiends is also well-equipped to defeat fey.
We must also consider that this paladin oath is ancient, and its ability to turn evil fey with the power of nature may have been much more relevant in the long-forgotten past. We can only assume from the relative scarcity of such creatures today that Oath of the Ancients paladins were successful.