The role that Tom played in Gatsby's death was that he was pretty much the instigator. Of course, it was not Tom who actually killed Gatsby. Instead, it was George Wilson who actually pulled the trigger.
Tom was involved, however, because it was he who led Wilson to Gatsby. He led Wilson to kill Gatsby by letting him think that it was Gatsby who ran Myrtle over with the car. He told Wilson this when Wilson showed up at Tom's house with a gun.
"[Wilson] came to the door .... He was crazy enough to kill me .... His hand was on a revolver in his pocket every minute .... What if I did tell him? ... [Gatsby] ran over Myrtle like you'd run over a dog and never even stopped his car." [...] They were careless people, Tom and Daisy--they smashed up things and creatures and then retreated back into their money or their vast carelessness.... (Chapter 9)
Tom is able to convince Wilson that it was Gatsby who owned and drove the car that killed Myrtle, then kept going, so he is implicated in Gatsby's death because of inciting Wilson. He did nothing to try to talk Wilson down from killing Gatsby, though he knew Wilson had a gun and "was [acting] crazy..." As he unremorsefully tells Nick later, speaking "defiantly": "That fellow [Gatsby] had it coming to him. He threw dust into your eyes just like he did in Daisy's, but he was a tough one."
Nick can't forgive Tom, even though he sees that what he did was, to Tom, "entirely justified." To Tom, Gatsby was a gangster and a man trying to steal his wife. But on a deeper level, Tom, a racist through and through, never thought Gatsby--the former, possibly Jewish Gatz--was worth anything. His death would hardly register as a loss with Tom because Gatsby didn't, in Tom's eyes, qualify as a "Nordic."
"This idea is that we're Nordics. I am, and you are and you are and ... --and we've produced all the things that go to make civilization--oh, science and art and all that. Do you see?"
Gatsby was a social climber who didn't belong in Tom's Nordic, horsey set; Gatsby was the guy who should have been entering at the back door.