One of the things I have always made clear when arguing against discrimination by cake bakers, florists, etc. against LGBT people is that there’s a clear difference between a product issue and a customer issue.
A customer issue is when a business offers a good or a service to some customers, but refuses to offer the exact same goods or services to other customers because of sexual orientation (or race, or ethnicity, or sex, or whatever). Example — a baker will bake a wedding cake for an opposite sex couple, but refuses to bake a cake of exactly the same design for a same sex couple.
A product issue is when a business refuses to offer a particular good or service for anyone, regardless of who they are. Example — a kosher deli refuses to sell a ham sandwich to a Baptist who asks for one, because they don’t sell ham sandwiches to anyone.
This can get a bit tricky when we’re talking about a generic product (such as a cake or a t-shirt), but the refusal comes down to a particular design or customization of that product. Still, as long as a business would deny a particular design or printing for ALL customers, they should be able to deny that design or printing for any customer. However, if a baker refuses to do a rainbow motif for a cake for a same sex couple, the baker must refuse to do rainbow motifs for everyone — even a child’s birthday cake.
If the business has advertised “we will print anything”, they must live up to that. However, no business is likely to offer that, as this would obligate them to print swastikas, racial slurs, pornographic images, or other designs or writing they may find objectionable.
As long as this t-shirt shop would not print t-shirts for any customer advertising an LGBT event, and as long as they never promised “we will print anything”, they are within their rights to refuse, as this is a product issue.
Personally, I think it’s an unwise course of action, as they are certain to get lots and lots of bad Yelp reviews. And the publicity surrounding this case probably hasn’t helped them.