On most of the the Greek Islands water is a problem to a greater or lesser extent. Any water, either hotel room tap, restaurant or bar or what have you may be suspect. On an island you are much more likely to drink imported from off island bottled water of which there is usually an abundant supply at hand.
Since on many islands you can drink the tap water its good to know that In fancy joints, some unscrupulous waiters or waitresses may bring bottled water as a matter of course unless you specify Tap water and ask for a 'carafe' and 'potiria' (glasses). I always try to eat where the Greeks are eating and not the tourists traps.
Considering that the latter are by law brought with no extra charge, and taking note of the fact that plastic bottles constitute a huge percentage of the plastic garbage in Greece (often burned on hillsides, and in any case, not bio-degradable), visitors to Greece would do well to ask if the public water is safe to drink before buying bottled water, and to carry a thermos or water bottle with them to fill up where there are taps with spring water.
In towns where there are such taps, eating establishments often stock up on this water for those who ask for the carafe rather than the bottled kind. All of this does not in any way deny the fact of occasional unsafe, bad tasting, or over chlorinated public water usually from a big city tap.
As with everything in Greece, it is good to ask a few local people for their opinion. Ask the people in the tourist offices if they drink the local water (since they have no stake in seeing you buy bottled water), and also if there is local spring water in taps around the town. Islands too are often short on water. In Syros for example the tap water is not potable. Always ask, but even so Montezuma has yet to visit his revenge upon Greece. Barbarossa yes, Montezuma no.