A lot of NCAA tournament games are officially hosted by a school in an arena that they call home. NCAA rules prohibit teams from playing in arenas where they have played four or more home games though (Final Four excepted, of course), which is why Syracuse can't play in the East Regional this year.
So while teams can't play games in their home gyms, they can play in their home states. Here's some information I've compiled about teams playing in their home states. It dates back to the beginning of the 64 team field in 1985.
one two teams have ever played in the Final Four on home state turf: 1994 Duke, which beat Florida but lost to Arkansas in Charlotte, and 2009 Michigan State, which beat UConn but lost to North Carolina in Detroit.
The Final Four is in Indianapolis this year, officially hosted by Butler and the Horizon League. With Purdue's Robbie Hummel going down, it looks like the Blue Devils' and Spartans' place in history is secure unless Butler catches fire in the tourney.
Fourteen teams have played with the Final Four on the line in their home state. Two were Cinderellas, and both in 1990: 10 seed Texas lost to 4 seed Arkansas in Dallas, and 11 seed Loyola Marymount lost to 1 seed UNLV in Oakland. Only two more were underdogs: 3 seed Syracuse beat 1 seed Oklahoma in Albany in 2003, while 2 seed Texas lost to 1 seed Memphis in Houston in 2008. The lesson is this: don't bet on a home state underdog in the Elite Eight unless it has Carmelo Anthony. Or something like that.
Of the remaining 10 teams that were favorites, only one fell. It was 2001 Stanford, which as a 1 seed lost to 3 seed Maryland in Anaheim. It is also the only team to lose as a 1 seed in its home state.
Twenty-one teams have played in the Sweet 16 in their home states. Underdogs are a respectable 3-5 in these contests. The afforementioned Texas and Loyola Marymount teams from 1990 hold two of the three wins, knocking off 6 seed Xavier and 7 seed Alabama, respectively. The third win came from 1994 Florida, which as a 3 seed beat 2 seed Syracuse in Miami.
Of the 13 favorites playing in the Sweet 16 in their home states, just two have fallen. In 1986, 2 seed Georgia Tech lost in its home town of Atlanta to 11 seed and eventual Final Four team LSU, while in 2002, 8 seed UCLA lost to 12 seed Missouri in San Jose.
First round losses by home state favorites are rare, largely because of the selection committee's bias towards giving high seeds games close to home when possible. Only four home state favorites (in 59 games) have lost in the first round: 1985 Purdue (6 seed, lost to Auburn in South Bend), 1985 Texas Tech (6 seed, lost to BC in Houston), 2002 USC (4 seed, lost to UNC-Wilmington in Sacramento), and 2009 Ohio State (8 seed, lost to Siena in Dayton).
Second round home state upsets are largely the domain of 2 seeds for some reason. Of the seven favorites to lose second round games in their home states (in 53 games), six were 2 seeds with the seventh being a 4 seed.
Here's your information overload for the day. These tables show records of the various seeds in various situations. The titles explain them.
Seeds' Performance in Home State
Seeds' Performance Against Teams Playing in Home State
Performance in Home State Versus a Weaker Seeded Team
Performance in Home State Versus a Stronger Seeded Team