Legal expert Eugene Volokh:

That’s the question brewing in Lexington, Kentucky. The Gay and Lesbian Services Organization has filed the following complaint with the Lexington Human Rights Commission.

... even if the Ordinance does prohibit what Hands On Originals did ... then the Ordinance unconstitutionally compels speech, because it requires printers to print material that they do not want to print. Abood v. Detroit Bd. of Ed. (1977) and Keller v. State Bar (1990) reaffirmed that the government generally may not compel someone to give over money to a private or even quasi-public entity when that money will be used for political or ideological speech. (There is an exception for when the government is acting as employer or regulator of the bar, and the compulsory payments are germane to the collective bargaining functions of a union or a bar association, but that does not apply here.) If so, then requiring someone to actually physically print political or ideological speech is an even clearer First Amendment violation.

Indeed, speech on T-shirts is as protected as speech in books. Under the GLSO’s view, a book publisher that is opposed to (say) Scientology could be required to print pro-Scientology books. Likewise, a printer that hates Nazi ideology could be required to print pro-Nazi leaflets in those jurisdictions — such as Washington, D.C. and Seattle — that ban public accommodations discrimination based on political affiliation. That, it seems to me, can’t be constitutional: Though the publishers (or the T-shirt printer) would be required to produce speech, rather than utter or display it himself, the creation of speech is itself speech, and compelled creation of speech is a speech compulsion.

... Any printer, whether religious or not, has a First Amendment right to choose what messages it will print and what messages it won’t print.