I don't know if I can provide a good answer to the question posed by the title. On one hand, voting is certainly one of the rights and responsibilities of citizenship, and citizenship only. On the other, it doesn't seem crazy to me think that people who pay taxes and live in a city should be able to have a say in how those taxes are used in that city.
One cannot stop there however; there also other complications inherent in the question. One side of the debate can make claims, in many cases justified, that allowing non-citizens to vote is really just a shameless vote grab. It is also fair to ask where the freedom to vote will end. In state-wide elections? Nationally?
The other side, again, in some cases justified, can claim vaguely nationalist or racist undertones in the rhetoric of those who are against the vote for non-citizens. Who has had the right to vote has also changed over time, that side would rightly say, and in saying so could point to times when property ownership, race, and sex were more important in the law's eyes than equality. Perhaps this is just the next step in that, to date, virtuous cycle they might say.
There are also broader questions on immigration policy, states rights issues (could states be forced to allow non-citizens to vote?) and fairness questions implicated in this conversation. One could even go to the other end of the spectrum and ask whether those who, in many cases, don't take advantage of the rights they do have to vote, have the right to tell others they cannot. All of these questions and topics are relevant, and this happens to be one of those situations where I can see some sense in just about all of them.
In any case, with the recent news that New York City is considering allowing non-US citizens to vote in local elections, it should be expected that this debate will be making its way to the forefront on the nightly news entertainment programs very soon. It should also be expected that the talking points about the topic will be expressed a bit more forcefully, unobjectively and venom-filled than they are above. I think it is safe to say that no one, citizen or not, will benefit from that...