While I did enjoy watching The Big Lebowski, I know for a fact that I didn't understand it completely by any means. So rather than attempting to delve into the philosophical or psychological aspects of the film, I'm going to focus simply on the relationship between Walter and Donny.
Donny serves as Walter's lap-dog. Almost any time Donny attempts to enter a conversation between Dude and Walter, he is greeted by a "shut the f--- up Donny!", courtesy of Walter. There are only a few times in the movie when Walter tolerates Donny, the most notable one being at the bowling alley bar, when the Dude and Walter have gotten into an argument. Frustrated with his best friend and bowling partner, Walter decides to instead take Donny along to bowl. Donny, thus, seems to be a sort of safety option/friend for Walter when his relationship with Dude has its troubles.
It's probably far-fetched, but I think that after Donny's death, Walter realizes that he no longer has his safety "friend." Perhaps deep-down Walter is aware of the fact that he tends to do things that, however well-intentioned they are, get on Dude's nerves. Knowing that Donny will no longer be there is a scary thought for Walter.