In 1976, while an undergraduate at Ithaca College, Burch and his brother Robert started Eagle's Eye apparel with a $2,000 investment, buying sweaters for $10 each and selling them for $15. Burch decided to find a factory to produce a new brand of "preppy" sweaters so that he could sell them door-to-door on campus. His operation soon expanded to other campuses, and, eventually, to Retail stores. Over the next decade, the company expanded to $140 million in sales and a national distribution footprint, including more than 50 of its own Retail stores. The company was partially sold to Swire Group, in 1989, and entirely, in 1998 in a deal that valued the brand at $60 million.

In 1982, Burch married Susan Cole in an Episcopalian ceremony. They had three daughters: Alexandra "Pookie" (b. 1985), Elizabeth "Izzie" (b. 1986), and Louisa "Weezie"(b. 1989). In 1996, he married the fashion designer Tory Robinson with whom he has three sons: twins Henry and Nicholas (b. 1997) and Sawyer (b. 2001). They divorced in 2007.

In 1993, Burch was a Producer for the feature-length romantic comedy film Watch It, directed by Tom Flynn, which received a favorable review from the New York Times. He was one of the earliest Investors in the Internet IPO for Internet Capital Group, and has made several Investments in telecommunications, including Aliph and Powermat. He was also an investor in the bottled water company, Voss Water, and the online office retailer Poppin. In a partnership with James McBride, he acquired Nihiwatu Resort on the Indonesian island of Sumba in 2013. After a $30 million investment, the pair opened the resort in 2015 under the name Nihi. In 2014, he invested in BaubleBar, leading an investment round of $10 million. That year he also partnered with Ellen DeGeneres to launch her lifestyles collection E.D.

In 2004, Burch partnered with Architect Philippe Stark and hotelier Alan Faena to develop the Faena Hotel+Universe in Buenos Aires. The investment into the building, first built in 1902 as a grain storage facility, was in excess of $100 million. In 2014 he once again invested in the real estate developer's projects after purchasing a condo in Faena House, a South Beach development.

In 2005, Burch purchased a luxury home in Southampton, New York for $14 million, and sold the home for $25 million four months later after renovating the property. Burch then founded J.B. Christopher, a supplier of construction materials to real estate developers, in 2006. In 2011, Burch and his partner Austin Hearst completed the development of a $19 million luxury home developments in Nantucket, Massachusetts. In 2012 he acquired a townhouse in the West Village for $11 million after he had been previously renting it. He subsequently undertook a renovation project that was left unfinished before the property was once again put on the market.

Burch was on the board of the Rothman Institute Orthopedic Foundation. He is a past President of The Pierre Hotel Co-op Board, and a past board member of Guggenheim Partners, and The Continuum Group. In 2013, Burch donated $1.3 million to his alma mater the Tilton School, and was on the school's Board of Trustees from 1982-1985.