Lennie is primarily a burden on George, which we learn in the opening scene of Of Mice and Men.
George holds Lennie's work card so that Lennie does not lose it. George provides the beans that the two men will eat for dinner (and Lennie whines about not having ketchup to go with the beans). And George is called upon to tell the story about the rabbits before the pair fall asleep. Many more examples exist in the story as to how Lennie is a burden on George.
Lennie also requires the patience of Slim and Crooks, in particular, as Lennie wants to play with Slim's pups when they are too young to be taken from their mother and as Lennie invades Crooks' privacy when George and the men have gone into town.
Lennie should not be seen as only a burden, however, as George, Crooks, and even Curley's wife find some companionship with Lennie.
Lenin is represented as a burden due to his addiction to ketchup. Throughout the novel, Steinbecks shows George's inability to satisfy Lennie's outrageous and unrealistic needs - particularly Lennie's desire for ketchup to go with his beans. One of the reasons perhaps why George is unable to afford some Heinz ketchup is because the Great Depression closed down many businesses.
To conclude, George shoots Lennie in the back of the head because he realises that his quest to find some ketchup for Lennie prevents him from reaching his own American Dream - a place where he can live of the fatta the lan.