The spell find familiar conjures a celestial, fey, or fiend spirit that takes the form of a beast. Do these special beings need food, drink, and sleep? If so, how much of it?
Creatures that do not need sustenance have that called out in their stat descriptions. The Undead Nature trait for skeletons, zombies, etc., grants the ability to survive without food.
A familiar “has the statistics of the chosen form,” and all of the listed forms are natural animals that would need to eat.
That is, if we are to assume that the statistics that a familiar gains from its form include traits such as a bat’s Echolocation and Keen Hearing then we also have to take a zombie's Undead Nature trait is a statistic — and that is a trait no familiar form has.
The familiar has the type of celestial, fiend, or fey, but these types (listed on MM p. 6) make no mention of not needing sustenance.
It’s another matter when the familiar is dismissed — then it is arguably not in animal form, and would just be a spirit that might not need to eat.
Well, do spirits eat or not?
There are no statistics for a “spirit” creature in the official rules. The only description I’ve found of a spirit comes from Unearthed Arcana’s Spirit Totem druid power: “The spirit…counts as neither a creature nor an object.” If we were to (1) accept this description from the unofficial UA rules, and (2) assume familiar spirits are similar to this one, then not being creatures, they would not need to eat like one.
Otherwise, whatever a spirit needs would be up to the DM. I would suggest it wouldn’t be much, since the spell says you can summon a dismissed spirit again, without listing any limitations or conditions.
Yes, but not all the time.
While present in the Prime Material (or whatever plane of existance the familiar is) it will have all the needs of the beast it is emulating. But also has all the capabilities of the beast it is emulating. It is almost impossible for a tiny animal to starve in a medieval town. It can eat your crumbs, forage, hunt just as any other beast of its type would. Also its dietary requirements are minimal.
While dismissed, the familiar is in a pocket dimension. We can assume this pocket dimension can sustain all the familiar needs (whatever they are) because there is no time limit it can remain dismissed without ill effects (it would be part of the spell description).
So for small periods of time in the active service of the spellcaster, I don't see why it would starve. From the previous paragraph, it can refresh itself after dismissed.
Now one last thought. It is a wonderful roleplay opportunity to feed and tend to your familiar. How the player treat its familiar can work to develop the character personality, and adds liveliness and flavour to the game. While at the market shopping, tell the DM you are looking for bird seed and a few mice for your owl. Or hire that street urchin to get some (live) mice for you.