Very little as it turns out.
Interesting bit from Caulton Tudor who says UNC's greatest concern is not the injuries but the fact there is universal agreement they will win the national title. Tudor then proceeds to name of countless times where the reverse of conventional wisdom happened such as NCSU winning in 1983 complete with gratuitous Jim Valvano quotes.
My initial response is to ask what control has UNC over any of this? The Heels return everyone from a 36-3 team and all the other teams lost something. UNC is the most experienced and arguably the most talented team in the country. Conventional wisdom, such as it is, will make them the favorite. This is not something they can control. They only thing UNC players and coaches can control is how they react to the hype. Since this group of players have been through this process of being #1 for two seasons now I hardly think that is a concern.
Seconday to that, I think Tudor misses the point and that is even if you declare a team to be a prohibitive favorite to win the NCAA Tournament, you are still talking about less than a 50% chance of succeeding. This was a point raised in the comments recently: What does being a leading favorite to win the national title actually mean in true odds? What it means is you have somewhere in the range of a 20-40% chance of winning an NCAA title. Before the NCAA Tournament, Ken Pomeroy usually offers up chances of how far each team in the tournament will go. In 2007, here are the teams he gave the best chance to win the national title.
Ohio State 11%
Now this was at the start of the tournament. As it turned out UNC and Kansas never made the Final Four and the team with the fourth best chance at winning, Florida, actually won the title. In 2008, Pomeroy gave Kansas a 36% chance of winning the title which proved correct. UNC and Duke were both given 4% chances of winning the title. Memphis and UCLA were at 13%.
My point is that what Tudor suggest is not earth shattering. In fact, on the basis of Pomeroy's analysis which is at the start of the NCAA Tournament and deals with more known data than the preseason prognistications do, the odds are greater someone else will win the title than the team everyone predicts. That is ultimately why we watch the games outside our desire to root for our favorite team. The fact the best teams in the polls walk into the NCAA Tournament with as much chance of winning as an elite baseball player has of getting a hit gives March Madness it's notoriety. Being a favorite in college basketball means you have better odds of winning that everyone else. The problem is the odds someone else out of the pool of 300+ schools will win the title is twice as good as the odds given to the favorite.
Such is the nature of the beast.