No, they won't have to. I'll speak to this from the perspective of the Catholic Church, and how they'll avoid having to perform same sex marriages.
The Catholic Church can already deny anyone a church/religious ceremony. Suppose I'm a Catholic and was married within the Church and later divorced. If I don't receive an annulment, the Church will not allow me to remarry within the Church. I can still go elsewhere, but the Church will not perform the marriage or acknowledge it as valid (a later annulment can get the Church to change its stance on this marriage).
Similarly, the Church (in the US) typically requires 6 months of marriage preparation. If you refuse to go through it, they will deny you a marriage within the Church (a friend of mine went through this last year, they were impatient and couldn't wait the extra 4 months apparently).
With regards to same-sex marriage, the Church's stance will not change (certainly not in our lifetimes), they'll continue to view same-sex relationships as sinful. Consequently they cannot, from their perspective, sanctify a marriage of that sort. This is not, fundamentally, different from the other reasons they use to deny people a Church marriage.
Edit: Regarding your second paragraph, yeah, I hear that a lot. But they're ignoring the cases where the Church already denies marriages. I already had this discussion with a coworker who happens to be a Catholic deacon this morning. It's frustrating that so many Catholics (and others) don't realize that they've already got the freedom to deny anyone access to the sacraments. I can be refused communion by a priest if he knows that I'm an (unrepentant) philanderer or drug dealer or murderer or that I've been excommunicated by a bishop. They are not obligated to perform baptisms just because someone asks if those asking aren't willing to go through the steps and agree to the things the Church requires. They won't ordain me just because I walk up to a bishop in a cathedral and ask to be ordained. A church marriage is a similar sacrament that the Church has laid out requirements for, and refusing to comply with those means you won't get married in the Church.
The Catholic Church can already deny anyone a church/religious ceremony. Suppose I'm a Catholic and was married within the Church and later divorced. If I don't receive an annulment, the Church will not allow me to remarry within the Church. I can still go elsewhere, but the Church will not perform the marriage or acknowledge it as valid (a later annulment can get the Church to change its stance on this marriage).
Similarly, the Church (in the US) typically requires 6 months of marriage preparation. If you refuse to go through it, they will deny you a marriage within the Church (a friend of mine went through this last year, they were impatient and couldn't wait the extra 4 months apparently).
With regards to same-sex marriage, the Church's stance will not change (certainly not in our lifetimes), they'll continue to view same-sex relationships as sinful. Consequently they cannot, from their perspective, sanctify a marriage of that sort. This is not, fundamentally, different from the other reasons they use to deny people a Church marriage.
Edit: Regarding your second paragraph, yeah, I hear that a lot. But they're ignoring the cases where the Church already denies marriages. I already had this discussion with a coworker who happens to be a Catholic deacon this morning. It's frustrating that so many Catholics (and others) don't realize that they've already got the freedom to deny anyone access to the sacraments. I can be refused communion by a priest if he knows that I'm an (unrepentant) philanderer or drug dealer or murderer or that I've been excommunicated by a bishop. They are not obligated to perform baptisms just because someone asks if those asking aren't willing to go through the steps and agree to the things the Church requires. They won't ordain me just because I walk up to a bishop in a cathedral and ask to be ordained. A church marriage is a similar sacrament that the Church has laid out requirements for, and refusing to comply with those means you won't get married in the Church.