I think that Lennie could be seen as a burden in a couple of ways. The first would be his absolute dependence on George. To a large extent, the dialogue between them at the start of the narrative reveals much in way of how much of a burden Lennie is to George. George is right to a great extent when he says that he could have gone farther if he did not have to constantly look out for Lennie. In this, Lennie's state of being a burden is revealed, albeit with a sense of cruelty in the depiction. Lennie needs someone to take care of him, something to which Aunt Clara knew enough to make George promise. Lennie needs someone to guide him, tend to him, watch over him, and to ensure that he is not manipulated by others. George learns this when he tells Lennie to jump in the lake as a joke, for fun, but something that Lennie does with severe intent. In this, George realizes that Lennie's burden is not only his limited intelligence, but also that he is susceptible to doing things for others that might not be in his best interests. It is here where Lennie can be seen as a burden or a weight to those who are around him. I don't think that this is the reason that George does what he does at the end. I don't think that the ending of the novel is undertaken by George as a way to free him. However, one of the reasons why George needs some steadying at the end of the novel from Slim is because he is free of a burden that had been around him from the start of the narrative. The instant alleviation of this burden is what causes a sense of shock in George, something he himself has a difficult time in believing.
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.