The Sheboygan Redskins played in the National Basketball League from 1938 until 1949, when the NBL merged with the Basketball Association of America to form the NBA. The Redskins wore sleeved jerseys from 1940 until around 1942, but by the mid '40s had switched to the standard sleeveless jersey, which they wore during their only season in the NBA.
The Boston Celtics wore sleeved jerseys for their inaugural year in 1946-47 - these lasted a single year before switching to a sleeveless jersey that, with a few minor changes, endures to this day. With rumours flying around that more teams will be introducing sleeved jerseys in upcoming seasons, it might be nice to see the Celtics break these unis out again.
Finally for now, we come to the St. Louis Bombers. Whilst the Bombers used standard jerseys from 1946-49, for their first (and only) season in the NBA they deployed a pair of sleeved jerseys which, it has to be said, look more like cheap T-shirts (great for a DIY project!). Interestingly, the Bombers used the same red shorts both at home and on the road.
So there we have it - the Warriors might be the first modern NBA team to wear a sleeved jersey, but they're not the first overall - the St. Louis Bombers has them beat by 64 years! It remains to be seen whether sleeves will catch on - although with the twin behemoths of Adidas and the NBA behind it I wouldn't rule it out.
Added March 18, 2013: Posted today in the Uni Watch ticker -- "Major, major find by Paul Dillon, who came across a photo of the Knicks playing the Baltimore Bullets in 1954, with the Bullets wearing sleeves! This contradicts the recently established narrative of the Warriors being the first NBA team to wear sleeves (yes, the Celtics wore sleeves in 1946-47, but that was before the NBA’s founding). It also shows once again how little we know about NBA uni history."
So it appears that the Bullets in 1954 may be the most recent team to wear sleeves in the NBA until the Warriors donned them this season.