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.
the familiar has the statistics of the chosen form, though it is a celestial, fey, or fiend (your choice) instead of a beast.
The Statistics, or stat block, of a monster starts with Size (MM6). Descriptions that come before size are thus not part of its Statistics. While there are several monsters with Undead Nature or Construct Nature that say they do no need to eat, these traits are listed before the stat block and are not, properly speaking, part of their Statistics (and with the release of MP:MotM these have been standardized to Unusual Nature).
Thus, whether or not an animal eats is not part of its Statistics. A spirit can assume the "form" of an animal, including all of its statistics, without necessarily assuming its need to eat (or lack thereof). Whether or not a spirit needs to eat is based on the nature of spirits, not based on the nature of the forms they assume.
Unfortunately, little is official about the nature of spirits and we are not told directly whether or not they need to eat. Until we have a full description of a spirit, including the lore before the stat block, we won't know what their "nature" is.
However, we can make some inferences from rules, and some conclusions that follow from how we actually play.
As an action, you can temporarily dismiss your familiar. It disappears into a pocket dimension where it awaits your summons.
The familiar may be dismissed into a "pocket dimension" and later recalled. No time limit is given for how long the familiar may remain in the pocket dimension, a place where it is not eating. If you play in a world where the familiar may exist for days in its pocket dimension before being called forth, then your DM is tacitly saying that spirits don't need to eat. If your DM requires that you call forth your familiar to feed it, then they are ruling that spirits do indeed need to eat.
In fact, the familiar might not even be breathing, since we know from the description of a bag of holding that the air inside that pocket dimension is finite.
Breathing creatures inside the bag can survive up to a number of minutes equal to 10 divided by the number of creatures (minimum 1 minute), after which time they begin to suffocate.
Other permanent extradimensional spaces (handy haversack, portable hole) have similar limits to breathing on creatures shut within them. Temporary extradimensional spaces (rope trick, magnificent mansion) don't, but perhaps their duration expires before the air would be used up, or perhaps they simply allow air to pass in their opening.
Regardless, if you play that a familiar can survive indefinitely in its pocket dimension, it seems reasonable that it does not need to breathe or eat.
Further, consider that familiars can be used as long-distance messengers by their casters. Would you, as a DM, allow a familiar to carry a message for their caster across a desert, ocean, frozen waste, or any other terrain in which there was limited access to fresh water? If so, and you insist that they need to drink, you are effectively limiting their travel range to four days, for after that they would be immobile from exhaustion. If you allow them to travel more than four days across a terrain without fresh water, you are saying that they don't need to drink.
Given that feeding and breathing are not part of the Statistics of their form, that familiars while in their pocket dimension (as usually played) do not need to eat or breathe, and that (as usually played) they can be sent great distances, it seems entirely possible that they do not need to eat, drink, or breathe at all, at least the way they are typically portrayed.