Get ready for a whole lot of DM Fiat
Nothing in the rules clearly states whether or not familiars need to eat, or need to sleep. We don't know much, RAW, about the nature of spirits. There is room for a DM to make decisions either way. A wise DM will make decisions that dovetail with the kind of game they want.
Eating and resting are not part of a creature's stat block
Answers to questions about familiars typically start with the following quote:
the familiar has the statistics of the chosen form, though it is a celestial, fey, or fiend (your choice) instead of a beast.
Unfortunately, many people then erroneously assume that since familiars share a stat block with the creatures whose forms they take, they take on the same needs for sleep or food as the creatures have. This is not necessarily correct. The Monster Manual (p.6) clearly states that the Statistics, or stat block, of a monster starts with Size. Any descriptions that come before size are thus, by definition, not part of its Statistics. While there are several monsters with Undead Nature or Construct Nature that say they do no need to eat or sleep, these traits are listed before the stat block and are not, properly speaking, part of their Statistics.
Thus, whether or not a beast or other creature 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. The rules do not tell us whether spirits need to eat or sleep - it might be that they do, but if they do, it is not because the creatures whose forms they wear do.
If, as the caster of a familiar, the game expected you to feed it, it seems reasonable that the rules for how to obtain that food and how much food it needed would be player-facing, as they clearly are for characters and mounts.
A character's needs for Food and Water are found in Chapter 8 of the PHB (Adventuring).
Characters can purchase food explicitly explicitly for themselves:
"Adventuring Gear", PHB 153
Rations. Rations consist of dry foods suitable for extended travel, including jerky, dried fruit, hardtack, and nuts.
Characters can also purchase food explicitly for their mounts:
"Tack, Harness, and Drawn Vehicles", PHB 157
Feed (per day) 5 cp 10 lb.
The 'feed' items will meet the needs of creatures from the 'Mounts and Other Animals' table, and the food ('rations') will meet the needs of characters - but what about the needs of familiars? A rat can likely eat the same things as a human, and a crow the same things as a mount...but what does a spider, octopus, sea horse, or bat eat?
Rather than purchased food, perhaps familiars need magically created food?
"Create Food and Water", PHB 229
You create 45 pounds of food and 30 gallons of water on the ground or in containers within range, enough to sustain up to fifteen humanoids or five steeds for 24 hours.
Again, the spell explicitly mentions characters and mounts (here called steeds), but not familiars. It does include pounds of food and gallons of water, so assuming that magic food is magically suitable for any body type, a character with a familiar could supply it with food - if any player-facing rules told them how much the familiar needed. But they don't.
Instead, we have DMG p.111, Food and Water table, which shows how much food and water are needed by different sizes of creature. So if a familiar is tiny, it might need "1/4 pound" of food, and "1/4 gallon" of water per day. On the same page, the DMG describes how much food and water can be obtained by foraging, making it possible to feed familiars that way as well.
Thus, there are no player-facing rules that explain how much to feed your familiar and how much that should cost. Why not? It could be because the Spirit Nature of familiars means they don't need to eat, but that is certainly not the only explanation (and I wouldn't argue that it is). It could also be because familiars need to eat, but this is not something the game deems important to track. Or, it could be that these rules are DM-facing because DMs are expected to decide for their game whether they want familiars to require food or not. We don't know which. What we do know is that the rules are not set up in such a way that makes it clear that familiars do need to eat.
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 (since it can't take objects with it) and not breathing (if its pocket dimension is anything like that of a bag of holding).
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.
Since a familiar can survive indefinitely in its pocket dimension, it seems reasonable that it does not need to breathe. If it does not need to breathe, perhaps it does not need to eat or sleep either - at least while in the pocket dimension.
DM, ask yourself...
If a wizard dismissed its familiar into a pocket dimension and left it there for several days, would you have it make Con checks for exhaustion from lack of food? Would you have it die from lack of water and have to be re-summoned (for 10gp and an hour of time)? If a wizard put its familiar on guard duty for several days straight would you have it gain exhaustion from lack of sleep? You already know, or you can quickly come up with, the answers to these questions for your own game. Given that the rules have nothing definitive to say about the answers, your own answers are more important, and are what will drive the resolution of the question Do familiars need to eat or sleep?
What I would do
At the start of each campaign, I ask myself a simple question: is this familiar a class feature or a recurring NPC?
In almost every game I have run, they are clearly a wizard class feature. They do exactly what the player wants them to do, using the rules from the spell, only. They are the means to an end for the player, and I am happy to run them as such, without the need for any tedious book-keeping about what they eat or when the last time they rested was. At best they get a cute name to keep track of what form they are in, but they don't even have personalities.
In one game I run, most familiars work as above, but one in particular is singled out because it is an imp. He has a distinct personality (as comic relief to give grief to the party paladin) and is an important source of information (because the party is currently in Avernus). I have invested in a distinct personality because it enhances the game; I have not worried about resource tracking because that would clearly not.
In one game I used to run, there were no familiars, but the Ranger had a beast companion which was a full-on NPC. It had its own likes and dislikes, preferred food, moods, and ways of interacting with the ranger and other members of the party. Had there been a familiar in that campaign, it would have gotten the same treatment. Why? Because that group of players cared very little for the mechanics of conflict and very much about role-playing relationships. Perhaps not co-incidentally, this group was my then-teenage daughter and her friends. They wanted to know how every character felt about every other character, and the animal companion had to have its own emotional life as well - they didn't much care about how many rages per long rest the barbarian got. How well the ranger kept the companion fed, and its response to that, was an important part of their playing experience.