Strictly speaking, unarmed strikes are not finesse weapons and cannot be used for Sneak Attack.
Instead of using a weapon to make a melee weapon attack, you can use an unarmed strike: a punch, kick, head‑butt, or similar forceful blow (none of which count as weapons).
So unarmed strikes don't count as weapons. Moreover, the Monk's Martial Arts ability says:
You can use Dexterity instead of Strength for the attack and damage rolls of your unarmed strikes and monk weapons.
This lets you use Dexterity instead of Strength, similar to the finesse property, but does not actually give the finesse ability. So even if unarmed strikes counted as weapons, they wouldn't count as finesse weapons.