What language is spoken in Athens, Greece?

To add to the other answers, and to answer a slightly different question 🙂 : between the 1300s and the 1800s, the region *around* Athens was substantially Albanian-speaking (Arvanitika). That’s why the map Brian Collins included in his answer has a patch of white. (A friend of mine once called that patch of white the αλβανότρυπρα, “the Albanian hole”.)

In fact, the village of Athens itself had two languages: Old Athenian, an archaic dialect of Modern Greek related to the dialect of Mani; and Arvanitika. Which is where the district of Plaka got its name from, from the Albanian pllakë, “old”: it’s the old town.

Any Jews living in Athens would have spoken Jewish Greek (Yevanic language). Romani would presumably also have been spoken—although speakers of the Agia Varvara variant of Romani, which is famous for having been studied linguistically [A Glossary of Greek Romany As Spoken in Agia Varvara (Athens)], are refugees from Turkey. The Muslims of Athens, I assume, spoke Turkish: I’m not aware of mass conversions of ethnic Greeks to Islam there, as had taken place in Crete.
 
Of course once Athens became the capital of Greece, both Old Athenian and Arvanitika were wiped out by the influx of speakers of what was to become Modern Standard Greek—a mixture of Peloponnesian and Katharevousa.