[RPG] Are there any official rules for what happens when a character goes without sleeping for a day or more

Nothing especially important in core is required to happen (you don't heal or recover spells), but there are optional rules to stay up with consequences.

You no longer are healing in core.

With a full night’s rest (8 hours of sleep or more), you recover 1 hit point per character level. Any significant interruption during your rest prevents you from healing that night.

If you undergo complete bed rest for an entire day and night, you recover twice your character level in hit points.

You can no longer recover spells in core.

To prepare her daily spells, a wizard must first sleep for 8 hours. The wizard does not have to slumber for every minute of the time, but she must refrain from movement, combat, spellcasting, skill use, conversation, or any other fairly demanding physical or mental task during the rest period. If her rest is interrupted, each interruption adds 1 hour to the total amount of time she has to rest in order to clear her mind, and she must have at least 1 hour of uninterrupted rest immediately prior to preparing her spells. If the character does not need to sleep for some reason, she still must have 8 hours of restful calm before preparing any spells.

There is an optional house rule for not sleeping.

The game has no rules for trying to stay awake through the night, writing down every word someone says without a mistake, or opening the stuck lid of a container without spilling a single drop of its contents. However, in the course of an adventure any of these situations could potentially make or break an encounter. You have to be ready to make up checks for such non-standard activities.

Using the example situations above, staying awake might be a Constitution check (DC 12, +4 for every previous night without sleep), with an elf character gaining a +2 bonus on her check because an elf is only giving up 4 hours of trance instead of 8 hours of sleep. (DMG 33)

So, by core, and if your DM doesn't care, you never need to sleep unless you need to recover spells or heal, but they may make optional rules or house rule it.