Walter Olson has a top-notch blog post over at Independent Gay Forum that describes why increased acceptance of same-sex marriage isn't going to lead to acceptance of polyamory.
I agree with all of his arguments and I'd add one. Gay marriage is, at the very worst, neutral for society while polyamory is pretty clearly harmful to society. The obvious harms of polyamory are likely to prevent its widespread acceptance.
The facts about gay marriage should come first. Now that we've had almost a decade of legal gay marriage, it seems reasonably safe to say that it has no detectable negative social impacts. If it did, say, harm heterosexual marriage or lead to more illegitimacy, that would be an argument against it. But there are no harms. Illegitimacy and divorce are both down from their highs; teen pregnancy is way down.
Meanwhile, only about 3.5 percent of the population identifies as gay indicating (no surprise) that increasing social acceptance of homosexuality is not exploding the number of people who identify as LGBT. Many people, me included, have come to support same-sex marriage because it's socially beneficial. In the long term, most of the impacts of gay marriage seem likely to be good: fewer people in intrinsically unstable "mixed orientation" marriages (leading to even less divorce), more loving two-parent homes for children, and more stable family environments for gay Americans. And these positive social externalities of gay marriage are likely to increase as gay marriage becomes more widespread.
On the other hand, a long social experience with polyamory indicates that the social results are awful. If they're patriarchal and primarily polygamous and limit the economic roles that women can take (as almost all known polygamous societies do) they will doom a lot of people to living in poverty. Self-described "fundamentalist Mormons" and the handful of backward Muslims that Olson mentions almost all live in poverty surviving off of government transfer payments and even crime. Polyamorous societies will, by definition, never have enough mates to go around. Always and everywhere, this has resulted in significant numbers of disaffected heterosexual males who have no hope of finding a mate.
And legitimizing polyamory would increase the number who practice it. Unlike being gay -- which, overwhelming evidence suggests, is not a choice -- polyamory clearly is. Its legitimacy would increase its prevalence.
If any major modern society ever moves towards legitimizing polyamory or anything like it, the social results are likely to be an unmitigated disaster in the short term. And this will create a very strong warning to anyone going down the same path. Gay marriage is increasingly accepted precisely because its results, to date, have been good for society. Polyamory on a large scale would have negative short-term results and that's a good reason to think it's just not going to happen.