Who pays for the travel for Heisman trophy nominees?

The only thing that can be said for certain is NOT the Heisman Trophy Trust, the organization presenting the award. They are considered a non-profit charitable organization and tax-exempt therefor can only give money to other tax-exempt organizations.

Other than that all parties involved are going to be making money... lots of money. Source for below examples.

The school may for the trip as the athlete is part their organization and I'm sure they can justify those expenses as "football related".

Baylor Univ. estimates that Robert Griffin III's Heisman Trophy win last year was worth $250M in "extra donations, increased ticket sales, licensing fees, sponsorship deals [and] an expanded deal with Fox Sports Southwest," according to Michele Steele of ESPN.com. ³

The athlete may pay their own way as they will likely be receiving a windfall by becoming a marketable commodity;

Pro Stars Ink President Ben Litvin estimates a Heisman winner "can pull down about $50,000 a year from three to four autograph sittings a year, three to four corporate appearances a year, two to three paid golf appearances per year, and three speeches" ³

Or it could be sports agents looking to do the athlete a "favor" in return for being selected to represent the player during their professional career.

The governing agency of college athletics said he [2005 Heisman winner Reggie Bush] and his family accepted huge amounts of money and perks from sports agents--including a limousine ride to the Heisman ceremony. ²