Just wondering, did they originate from one place, or did several cultures invent sleeves independently of one another? I imagine they were necessary in cold climates, so were they a foreign concept to, say, the pre-Columbian tropics?
The origin of sleeves comes from the first tunics.
The earliest garments were a single piece of cloth that was wrapped around the body, found in many ancient cultures around the world. Some are still worn today, such as the Indian sari (although it goes over a short tunic-like shirt). Tunics such as this one have been found in Egypt from over 3000 years ago. This tunic consisted of a length of cloth, folded in half, with a slit for the head and the sides sewn closed to create a sleeveless shirt. This was also the poncho worn in what's now South America.
In The middle east and Europe, the tunic developed sleeves. This was done by either by weaving the cloth narrower for the section that would cover the body or weaving a longer piece of cloth, cutting the end off, and re-attaching it as sleeves. Once the tunic was developed, it stayed largely unchanged for a couple thousand years. It was quite common to see these sleved garments in hotter climates of the Mediterranean and middle east.
Sleeves were not associated with cold climates, as the material was often a very light fabric. Pants, however, were only worn by people in northern climates until around the seventeenth century.
I'm not sure if you can say that any one culture invented the tunic, but the earliest surviving examples come from Egypt. Many other cultures also used the single wrapped piece of cloth to create 'sleeves' as part of their wrapped garment.
A good resource on early tunics and wrap clothing is Cut My Cote, by Dorothy Burnham.