Exactly what the author wants refuge from has to be read between the lines, given the way such writers twist their language into a hard-to-follow snarl in order to justify the underlying stance that they are the ones suffering from oppression when they can't discriminate and harass any more. Still, given recent battles in the culture wars, it's not too hard to figure out what this means:
Business owners and workers must be allowed to do their work and flourish without having to celebrate moral corruption and keep their mouths shut. Florists, bakers, photographers, and all small business owners have the right to refuse to participate in celebrations of things they find immoral, just as any sane baker would justly refuse to provide a cake for a Nazi event.
So, basically, a sanctuary city would be one in which anti-gay discrimination would remain permissible because the local authorities would not cooperate with federal laws and court rulings against it. Entire states have already tried this by refusing to accept last year's Supreme Court ruling on gay marriage, and the federal government made it clear that it would not tolerate this. Suppose a few cities here and there announced that they were becoming "sanctuaries" in which lunch counters and bus companies were free to discriminate against black customers again? How long do you think that would be allowed to stand? Where discrimination is concerned, the federal government is rightly uncompromising, because it has a duty to protect all its citizens even if they live in places where bigots form a local majority. Aside from the government, entities ranging from corporations to sports leagues have boycotted states which legalize anti-gay discrimination under the guise of "religious liberty", creating massive economic pressure. Such "sanctuaries" suffer heavy and sustained penalties.
The post also includes a lot of complaining about people having to home-school their kids to protect them from being exposed to messages of tolerance for those who are different. So presumably a sanctuary city would be one where public schools were officially tolerant of bullying of gay kids and perhaps even included some Christian doctrine as part of the curriculum. This might be easier to get away with given the power of local school boards to control local schools (the decentralization of control over education is one of this country's core problems, but that's a matter for another post), and easier still if it were done at private schools instead of public ones. But all that stuff basically happens already in many small towns where intolerant and ignorant religious people are in the majority. Aside from offering some local quasi-official legitimacy, it's hard to see what a "sanctuary city" would add. If they want open preaching of religious dogma as part of the curriculum in public schools with the support of local authorities, this has already been tried with creationism, and legal cases such as Kitzmiller v. Dover have shown that local authorities have no such power to ignore the Constitution. They are getting away with it in a few states where state authorities are complicit, but the Kitzmiller precedent suggests that determined lawsuits could end this.
The "sanctuary" concept is morally incoherent because the "oppression" these people claim to be fleeing from isn't really oppression. What they're trying to protect isn't their own equality and freedom, it's their ability to deny the equality and freedom of others (equality in the case of gays, freedom in the case of women who want contraception or abortion). The government may tacitly leave you alone to do your own thing, but not when "your own thing" is beating up on somebody else.
Something like what the post suggests did actually succeed for several generations in one case. When the Mormon Church finally yielded to government pressure and abandoned polygamy in 1890, a few small groups of hard-line dissenters left the Church and set up their own communities where they could continue polygamy. Known as "Mormon Fundamentalists" or FLDS, these groups built tiny, isolated settlements in remote areas, in some cases even across the borders in Mexico or Canada. These enclaves survived unbothered because they largely escaped notice. Only in the last few years, as a pattern of abuse of underage girls there entered public awareness, has the government begun to crack down on them. Again, the problem is that the "freedom" they were defending was the freedom to abuse children and keep women in subjugation.
Anyway, I don't think a tiny isolated compound in the middle of a desert is what Douthat or the readers of LifeSite News have in mind. They want modern conveniences and urbanity without modern challenges to the supremacy of religious dogmas, taboos, and prejudices. But that's fundamentally impossible, because of the exuberant free marketplace of ideas which modern popular culture creates. Consider the Middle East, a vast region of hundreds of millions of people where religion has been brutally dominant since the 12th century and where most governments actually support religious taboos and strictures. Its societies are in upheaval as millions question religious dogma or flatly (if quietly) abandon it, as women demand more freedom and education, as young people assert self-determination in sexuality and marriage. This is happening now, after centuries of stagnation under religion, because Western movies, TV, music, books, and the internet are flooding the region with new ideas and imagery. The Islamic backlash of political fundamentalism and even terrorism can kill people, but it can't stop these changes. What chance would Sanctuaryburg, Alabama have of keeping out the influences that would turn much of its own next generation against the culture it's trying to protect? FLDS enclaves ban TV and the internet. In a normal city, this would be impossible.
Muddled attempts at creating wingnut sanctuaries on the city or state level will doubtless persist for some time, but in the long run they will fizzle out and fade away.