There was no evidence that Elmo killed them, and there's no way they'd release a convicted murderer over one man's story. If they would, don't you think every prisoner would pay someone to make up a story like that?
Killing Tommy was like shooting a squirrel to the Warden and Captain Hadley. There was no downside to it as far as they were concerned and they had absolutely no compunction of killing anyone.
But if Tommy's story was even a tiny bit believable, it could have meant a retrial for Andy even if it didn't ultimately lead to an acquittal. If nothing else, it could have initiated an investigation or perhaps even a transfer for Andy out of Shawshank if the investigation showed some merit. Rocking the boat might overturn it.. For the warden to keep the status quo, it was critical for him to keep Andy at Shawshank for life.
As far as the Warden was concerned, snuffing Tommy simply and easily ended everything right there and then - and it would have, were it not for the little rock hammer and the hole under a movie poster... Until Andy escaped, Norton was right.
Agreed. Killing an inmate was probably something they would have like to avoid, but not too hard of a decision. And from their perspective, they felt it was necessary so they did it.
As long as Tommy was alive there was always the chance that the 'wrong' person might hear his story and reopen Andy's case. The Warden wasn't going to take that chance.
I always got the impression that the Warden took Tommy out because of the comment Andy made about not spilling the beans about the scams if he ever got out! Andy had too much dirt on him and maybe he feared blackmail, he knew Andy was a damn sight smarter. As long as Tommy was around Andy would be like a dog with a bone. No more Tommy, no more bone!
In the book Tommy was just transferred to another low security prison in exchange for his silence. I guess the scriptwriter had to spice things up a bit for the film.