The real reason for this is that man-killing means, sooner or later, the arrival of white men on elephants, with guns, and hundreds of brown men with gongs and rockets and torches. Then everybody in the jungle suffers. (1.19)
The animals make it seem like men are more apt to revenge than they are, but as we learn, the animals are just as hungry for revenge as man is.
"He shall not be killed. […] He shall hunt thee!" (1.39)
See? That didn't take long at all. A mere twenty paragraphs after our first quote, Mother Wolf declares that Mowgli will one day take revenge on Shere Khan.
"The time comes when this naked thing will make thee roar another tune, or I know nothing of Man." (1.63)
Bagheera tries to pin Mowgli's revenge on Shere Khan on Man's vengeful nature, but Mowgli would never have tried to kill Shere Khan had all the animals not constantly reminded him that he had to do it.
"I will surely come," said Mowgli, "and when I come it will be to lay out Shere Khan's hide upon the Council Rock." (1.145)
Here we see that Mowgli has drunk the animals' Kool-Aid about how it's his destiny to kill Shere Khan. The wolves want him to do it; Bagheera wants him to do it; and by this point, Shere Khan's big mouth has made Mowgli want to do it anyway.
Now a snake, especially a wary old python like Kaa, very seldom shows that he is angry, but Baloo and Bagheera could see the big swallowing-muscles on either side of Kaa's throat ripple and bulge. (3.73)
Kaa is plotting his revenge against the Bandar-log, and it's going to be fierce. They called him names, but it looks like he's going to swallow them whole. Words may never hurt you, but they may cause a giant snake to come out of the jungle and eat you.
"When [Shere Khan] returns he swears that he will lay thy bones in the Waingunga." (5.18)
Shere Khan just can't let it go. Even after Mowgli leaves the Wolf Pack, Shere Khan is determined to get the man-cub snack that escaped him years ago.
"Shere Khan could never fast, even for the sake of revenge." (5.46)
Here we see another example of Shere Khan's incompetence. Mowgli plans his revenge on Shere Khan in about ten minutes. Shere Khan has had months to plot his revenge against Mowgli, but he can't even keep himself in good physical shape by fasting in order to do it.
"Now, once more, Akela," [Mowgli] cried. "Bring the herd in." (5.99)
Mowgli gets Akela to herd the buffalo into the man's village to wreak some havoc as a little payback for kicking him out.
"Those who kill snakes get killed by snakes." (9.43)
I will remember what I was. I am sick of rope and chain. (11.1)
The verse at the beginning of "Toomai and Elephants" makes us think that one day the elephants might break free and exact revenge against their captors. But they don't. The gentle giants merely break free and dance.