The law is the highest authority
In a secular society with non-discrimination laws, all citizens must abide by those laws
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Context
Most Western liberal democracies organise themselves as secular nations. As such, the laws they write and the protections they assure their citizens are meant to uphold their liberties regardless of, and often in the face of, religious sentiments.
The Argument
The principle of non-discrimination is the building block of many developed nations. Every person, regardless of creed or race or gender, ought to be treated equally. For this purpose we institute laws that make it illegal to fire, hire or refuse service to people on the basis of who they are.
Given that we hold this principle of non-discrimination in high regard in both our courts of law and personal lives, it makes sense for these principles to be prioritised when we have such disputes. It is a poor state that watches helplessly as swathes of its population are mistreated.
These non-discrimination laws/principles should therefore be respected by everyone - regardless of religious affiliation.
Counter arguments
It could be argued that discrimination on the basis of religious beliefs is an equally important question in this discussion. If a Christian or a Muslim were to be refused service because they were people of faith, that would be unacceptable. It would also be unacceptable to force Christians or Muslims to feel that they are not equal citizens because of their religious beliefs.
Premises
P1. The law states that discrimination is unacceptable on any grounds
P2. It is legitimate for the law overrides religious teachings that claim discrimination of LGBT couples is necessary
C1. Therefore, religiously based discrimination against LGBT couples in unacceptable.