In The Dark Knight Returns 2, an elderly Green Arrow turns up but he's missing an arm.
What happened to him? Does Green Arrow lose an arm in canon somewhere or is it just for this particular story?
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The evidence is for it being due to Superman. In the The Dark Knight Returns comic, when Batman mentioned he was going up against Superman, Oliver rubbed his stump and said something like "It still hurts when it rains." That suggests that Superman was the source. That leads to the question of the circumstances.
The most likely case is that he lost it to Superman deciding to amputate it. In Green Arrow Volume 2 #100, released 9 years later:
Green Arrow ends up shot, with his hand stuck in a deadman's switch in the bomb containing the mutagen. As soon as he removes his hand, the bomb will detonate. Superman arrives on the scene .... On the plane, Superman goes over all the available options with Oliver. Oliver tells Superman to destroy the bomb and Oliver as well. Superman tells Oliver he sees one way out of this for Oliver, but it involves amputation. Oliver says he'd rather die and Superman tells Oliver the choice is not his.
In the baseline universe, Oliver refuses to let Superman do the amputation and he's blown up by the bomb, only to be later resurrected by Green Lantern by the bits of him stuck on Superman's cape. Within the The Dark Knight Returns continuity, instead, perhaps the amputation happened, making the bomb situation an explanation 9 years later. Alternately, we just have to assume that, at some point, Superman just decided to blow Oliver's arm off, possibly as part of the theme of Superman being the government's enforcer to prevent vigilantism.
It's alluded to in Dark Night part 1, when Kent and Bruce are talking about "retirement" due to goverment involvement. The choices were a) "stop" or b) work for the USA, Superman chose the later while the rest of the Justice League retired except Oliver, who didn't want to.
Superman had to "step in". How do you prevent an archer from shooting arrows? You take away his arm and you throw him in jail
The rest of us learned to cope. The rest of us recognized the danger... of the endless envy of those not blessed. Diana went back to her people. Hal went to the stars. And I have walked the razor's edge for so long... But you, Bruce... you, with your wild obsession...
They'll kill us if they can, Bruce. Every year they grow smaller. Every year they hate us more. We must not remind them that giants walk the earth.
You were the one they used against us, Bruce. The one who played it rough. When the noise started from the parents' groups and the sub-committee called us for questioning... you were the one who laughed... that scary laugh of yours. "Sure, we're criminals", you said. "We've always been criminals". "We have to be criminals".
We almost threw a party when you retired. By then the PBI was in it and the things were getting out of hand. And there was that trouble with Oliver. Do you remember why you retired, Bruce? No... just look at you... You'd do it again... and like a murderer, you'd cover it up again. Nothing matters to you... except your holy war.
They were considering their options and you were probably still laughing when we came to terms. I gave them my obedience and my invisibility. They gave me a license and let us live. No, I don't like it. But I get to save lives and the media stays quiet.
But now the storm is growing again... they'll hunt us down again... because of you.