Does a Construct need to sleep?
In the general case, no.
Does a homunculus need to sleep?
Probably not, but it's open to a DM ruling. It would be reasonable to rule that a Homunculus need not sleep, since it is a construct.
From MM pages 19 / 166: Constructed Nature (Animated Object / Golem)
"An animated Object (or Golem) doesn't require air, food, drink, or sleep."
MM page 188:
Homunculus, tiny Construct
A Homunculus is a construct."
MM page 6:
"Constructs are made, not born."
When we look at other constructs like Animated Objects and Golems, they don't sleep, so it's reasonable to rule that a Homunculus need not sleep and thus be a great alarm clock/alarm for a gnome who needs his beauty sleep.
Caveat:
Modrons, Animated Objects, and Golems are not presented as having immunity to sleep, nor are homunculi. A save versus sleep might be necessary were the spell cast upon them. ("Sleep" / "Asleep" are not conditions. The sleep spell creates the Unconscious condition).
Counter Argument: since the line "constructed nature" is not included in the description, as it is for other constructs, it isn't treated as a construct for that particular case of "constructed nature."
Recommended Ruling: it does not need sleep since it is a construct.
Put another way: you can put a Homunculus to sleep, but it doesn't need sleep. (No amount of beauty rest will make it any prettier. It's an ugly baby that only its creator could love ...)