Several times now, we've seen Barry travel through time.
The season 2 finale saw Barry travel back in time preventing the death of his mother.
Has there been evidence of things changing in Star(ling) City because of this?
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In the crossover episode, on Arrow's part of it, Barry traveled back in time because everybody died, and undid it. Since the team is still alive in Star City in present episodes, I'd say that serves as de facto evidence that Barry's time traveling in Central City affects Star City.
Similarly, Flash's first time travel was to stop a tidal wave destroying the city... since in episodes of Arrow subsequent to that, in Star City they don't act like Central City has been destroyed (and indeed still visit it), that's also evidence the time travel effects both cities... which stands to reason... it would in fact be extremely confusing if it didn't!
But while Barry's time travel does affect both cities, we probably will not often be shown it, particularly in cases where either the new timeline eventually will get reverted (as may be the case for the S2 finale), or where the old timeline is wiped out before we see the effects it would have had on Star City. So far I think the only case that doesn't fall into these categories (at least potentially) is earlier this season when Barry time travelled back to get information from Thawne. The only result we've seen so far from that, that I can recall, is Hartley (The Piper) who now has been an ally for some time. But since he never really played a role on Arrow, any impact on Star City would have been negligible.