Tsay was born in Ziguan District, Kaohsiung on Taiwan's southwestern coast. He attended National Cheng Kung University, where he received a bachelor's degree in 1971 and a master's degree in 1973. From 1975 to 1978, he was a lecturer in National Chung Hsing University's Civil Engineering Department. He then moved to the United States to pursue a Ph.D. at Cornell University, where he defended his doctoral dissertation on numerical solutions to certain hydraulics problems in 1982. Because of Tsay's support for the Tangwai movement, Taiwan's Kuomintang government blacklisted him, making it impossible for him to return. Instead, he pursued his academic career in the United States for the next eight years, and was naturalised as a U.S. citizen. After conducting postdoctoral research at Cornell University from 1982 to 1983, he took up a position as an assistant professor at Syracuse University, and was promoted to associate professor in 1986. He returned to Taiwan in 1990, a few years after the end of martial law on the island, to take up a new position as a professor at National Taiwan University.
In October 2008, Tsay led the Taiwan Association of University Professors in a hunger strike outside the Legislative Yuan building in support of relaxation of the procedures for public referenda. Since the passage of the 2003 Referendum Law, the government would only review referendum proposals supported by the signatures of at least 0.5% of the voter turnout in the most recent presidential election; whether or not to hold the referendum remained the decision of a government-appointed board. Tsay's goal was to make it easier for a Taiwan independence referendum to be proposed and conducted. He eventually created the Taiwan Referendum Alliance in support of this goal. What was initially intended to be a 24-hour protest ended up turning into a years-long event. By January 2010, he had been assessed fines of more than NT$800,000 relating to his occupation of public space during the sit-in protest. Lacking funds to pay the fines, Tsay and his supporters began soliciting donations outside the Presidential Office in Taipei, due to which he was arrested and had further fines assessed against him, in that case for alleged violations of the Assembly and Parade Act. The donations were largely in the form of coins; Tsay and his fellow protestors spent nine hours in the police station counting the reported three hundred and sixty thousand coins. In 2012, Tsay and the TRA also initiated a walking protest in support of clemency for former president Chen Shui-bian, who had been imprisoned after a conviction for corruption charges. In late February 2014, about a week before the 67th anniversary of the February 28 Incident, Tsay led a group of fellow Taiwan Referendum Alliance members in removing a statue of Sun Yat-sen from Tang De-jhang Memorial Park in Tainan, a park named for a local lawyer executed during the military crackdown which followed the February 28 Incident. Kuomintang politicians including Fan Chiang Tai-chi condemned the TRA's actions. During the March and April 2014Sunflower Student Movement, Tsay and the TRA also led sympathy protests outside of the Legislative Yuan. After the removal of protestors from the square in front of the Legislative Yuan, Tsay led further protests outside the Zhongzheng First Police Precinct station.
