How do Tom and Daisy treat Gatsby in a despicable manner?

In The Great Gatsby, first of all, if
by destruction you mean death, Wilson is responsible for Gatsby's death--he pulls the
trigger.  And Tom tells Wilson that Gatsby owns the car that hit Myrtle, so Tom plays a
part as well.  Furthermore, Daisy lets Gatsby take the blame and doesn't tell Tom, as
far as evidence in the novel suggests, that she was actually driving the car.  So she is
responsible, too.

In terms of his death, Gatsby is
responsible for his own death only in that he takes the blame for the accident for
Daisy.  His act of love and protection gets him killed, you could
say.

If you're asking about more than just the death, Daisy
says she loves Gatsby but then withdraws from him because, as she says, Gatsby asks too
much--he demands that she announce that she always loved him and never loved Tom, and
she refuses to do that. 

Tom slanders Gatsby with
speculation about his past, etc., in an effort to keep
Daisy.

Gatsby, if you're asking about more than just the
death, causes his own failure by dedicating himself to an illusion, and trying to
recapture a past that never was.