New York City Council Speaker Christine Quinn wants to kick Chick-fil-A out of New York because its CEO, Dan Cathy, opposes gay marriage. Accordingly, she informed the head of New York University (which leases space to the one Chick-fil-A restaurant in New York City) that "Chick-fil-A is not welcome in New York City as long as the company's president continues to uphold and promote his discriminatory views [...] I urge you to sever your relationship with the Chick-fil-A establishment that exists on your campus."
My guess is that the university will regard this letter more as an unstated threat than as a mere statement of the Speaker's opinion, since universities, vulnerable as they are to ad hoc government regulations and ordinances, are obligated to cultivate municipal officials' goodwill. As a rule, business owners are subject to municipal predation that can drive them out of business, and are thus forced them to ingratiate themselves with city officials. Universities can end up with an enrollment cap or lose lucrative eminent domain prerogatives if they annoy municipal higher-ups.
The government cannot pressure a private institution to terminate a contract with another private entity due to that entity's speech. This constitutes a violation of the First Amendment. For example, a government official cannot pressure a private institution to fire an employee for his speech. Moreover, impermissible government action can occur even in the absence of explicit threats or pressure: if the government merely reprimands a public employee for his speech, or censures a private citizen for his speech, some courts find a violation of the First Amendment.
Quinn's pressure comes in the wake of similar attacks on Chick-fil-A by other politicians. As I noted in The Washington Examiner,
[Boston's] mayor said he would block Chick-fil-A from opening a restaurant there because its CEO opposes gay marriage. [He has since retreated from this position]
Similarly, an alderman in Chicago has said he will block a zoning permit needed for a Chick-fil-A restaurant in Chicago because of its CEO's views.
Under the Supreme Court's Umbehr decision, cities cannot punish firms or withhold even discretionary benefits like zoning permits over their speech. The Supreme Court long ago ruled that firms have free speech rights in its rulings in favor of Consolidated Edison and the First National Bank.
Chick-fil-A has faced unusually few discrimination claims of any kind for a restaurant chain. There is no evidence that Chick-fil-A discriminates against gay patrons, and it has restaurants in many cities than ban anti-gay discrimination.
Actually, Chick-fil-A's case against Chicago is even stronger than the business whose free-speech rights were recognized in the Supreme Court's Umbehr decision. The First Amendment applies with even greater force when the speech restriction is imposed through regulatory decisions, like a zoning decision, rather than when it is tied to a government contract, as in the Umbehr case. Withholding regulatory approval is even less permissible since it doesn't involve the government's power of the purse.
Nor is there any illegal-discrimination rationale for banishing Chick-fil-A, even assuming a city official could just ban a supposedly discriminatory restaurant chain without any due process. Chick-fil-A has had far fewer discrimination claims filed against it than most restaurant chains of its size. Despite its operation in cities with ordinances banning sexual-orientation discrimination in employment and public accommodations, I cannot find any reported case of discrimination against gay customers.
Moreover, Cathy's statement is not linked to illegal discrimination against customers or employees. In this respect, Chick-fil-A is not nearly as culpable as the various corporations that filed amicus briefs for the University of Michigan in the Gratz v. Bollinger case, where the Supreme Court struck down a race-based college admissions policy. These companies explicitly supported discrimination on the basis of race, yet no one has ever attempted to ban them. Unfortunately, one needn't wonder why.