Shere Khan does not kill Baloo in Rudyard Kipling's The Jungle Book. Despite being portrayed as a villain throughout Kipling's books and their movie adaptations, Shere Khan kills a surprisingly small number of characters. If and when he kills someone, it isn't usually a main or even named character. The Second Jungle Book is a good example of this. It starts off with a story called "How Fear Came," which likely takes place sometime during the events that occur in The Jungle Book. In the story, Shere Khan appears at the watering hole and brags to the other animals that he has just killed an unnamed person (or possibly people) an hour earlier.
The 2016 film version of The Jungle Book portrays Shere Khan killing Akela, which does not happen in the books. In fact, Akela helps Mowgli kill Shere Khan in the original story. Shere Khan also kills Mowgli's parents in this film. The 1994 film version, for its part, depicts Shere Khan killing a fair number of humans. In this adaptation, Shere Khan kills Mowgli's father, an unnamed soldier, and a sergeant named Claibourne. The closest Shere Khan comes to killing Baloo is in the 2016 film, where the tiger and the bear have a fierce battle. Baloo is left too wounded to continue fighting but ultimately survives. In the 1994 film, Baloo is shot by soldiers and left for dead, but the English physician Dr. Julius Plumford is able to save Baloo's life. To date, Baloo remains alive and well.