Yes they should. Together with signs if the owner is vegetarian, respects the Sabbath and which party he votes. The owner should also sign a statement that he/she is ant racist, anti sexist, against animal experiments, not against feminism and believes in global warming. If one of those points on the checklist is missing, the shop will be blocked by politically correct "freedom police forces".
A business should be allowed to refuse service. But not unless they give fair warning to anyone looking at their business that some people will be sent right back out the door if they request service. If it's a right not to serve people it's a right not to have to be humiliated.
Any business owner who feels that strongly should feel strongly enough to put a sign up so gay people don't have to go in in the first place.
A business has no obligation to tell you that they don't support gay rights just as you have no obligation to tell them that you are gay. If you're so afraid of humiliation, research the place before you go. But since it was mentioned I might as well point out that there is no such thing as a right to not be humiliated.
I would also like to point out that there are some businesses that don't support gay rights, but still serve people who are gay.
That used to say "Whites Only" "No Colored", "No Irish", etc? Surely you must be joking. Not only should these businesses not put up the signs, they should not be allowed to. Businesses have no right to refuse service to someone for discriminatory reasons. Frankly, I find the idea disgusting.
If someone is a private business owner they can deny service to anyone the moment they walk through the door. Being required to put a sign up saying if you support gays is outrageous. If the owner doesn't want to serve a person because of their sexuality, race, religion, etc. then they can. They would be crucified in the media but that doesn't change the fact that they can.