Do any sects permit consensual open marriages?

Here's a better, more appropriate answer on my part. My answer pertained to polygamy, which of course is incorrect in this case - however:

This Wikipedia article on open marriages indicates the following, emphasis my own:

Some critics object to open marriages on the ground that open marriages violate religious principles. For example, open marriages contradict traditional Christian doctrine. Open marriages also violate the prohibition against adultery in the Ten Commandments. The Jewish religion hold similar values as Christianity in these matters. Muslims or Islamic teachings however allows for polygamous marriages. The perceived validity of these objections depends entirely on individual faith. Arguments about faith, faith and rationality, and belief systems lie outside the scope of this article.

People in open marriages tend not to be very religious. Jenks, in a review of the literature on swingers, writes:

"Bartell reported that the majority of his sample did not attend church regularly. Fully two thirds of the respondents in the Jenks' (1985b) study had no present religious identification. This finding also is consistent with other studies. Gilmartin's (1975) figure for the swingers was 63%. When asked if they had been raised in a religious home over 68% said yes. Although a little over 70% said they did not currently attend church services in a typical month, the most frequent response concerning church attendance when growing up was every week. Thus, swingers were raised in religious home but, somewhere along the path to adulthood, a majority gave up their religion." (Jenks, 1998)

Christianity, for the most of it, defines marriage as one man, one woman in a covenant union for life. We see examples of how detrimental an open marriages can be in the case of many Old Testament figures (i.e. Solomon, David, Abraham, and more), though this case was more to polygamy than an "open" marriage. In many of these cases (though again, polygamous) the relationship pulled the person involved (like Solomon) away from the Lord. True, they repented, but the fact remains most, if not all, denominations consider open marriages as a form of adultery, and therefor a violation of the sixth commandment. This includes Catholicism, Lutheranism, and Baptists. Trying to research the subject will bring more results on Church v. LGBT issues, but not every church is the same.

I am unable to find reference to any official sects or doctrines which permit open-relationships (as distinct from polygamy and various forms of plural marriage). However, there certainly are a minority of Christians who practice various forms of open-relationships. The Liberated Christians web site^* has a section about various forms of open relationships within a Christian context. As far as I know, it's a completely amateur site, run by volunteers, and without the backing of any official church organization, denomination, or sect.

An overview of their beliefs is explained here, and says, in part:

The ancient Hebrews, as portrayed in the Old Testament, clearly believed in multiple partnerships and this practice is nowhere condemned by God.

When the New Testament scriptures are viewed as a whole, taking into account the cultural context in which they were written, it is clear that neither Jesus nor the writers of the New Testament condemned such practice, although it appears that polygamy had, for non-religious reasons, substantially declined within Jewish culture by the time of Christ.

Despite this biblical evidence, the Christian church has persistently opposed polygamous relationships and has, at times, actively persecuted families which chose to practice this lifestyle. The church has also used twisted interpretations of various scriptures in defense of its opposition to this lifestyle.

And they describe those who agree with them as:

Many of them are mainstream, evangelical Christians, including pastors and their wives, church elders and Sunday school teachers. Some are seminary-trained theologians. Others are lay men and women from many different church backgrounds or no organized Christian background at all. They come from all walks of life, including college students, doctors, ministry professionals, housewives and blue-collar workers. They may be your neighbors, co-workers or even your relatives. (Scary, isn't it?!) They are ordinary people who have simply acknowledged that they have polyamorous desires, as do most other people who are unable to admit it to themselves, their partners, their church or their God.

So not really a sect, per-se, but there are a number of Christians purported to share their views.

^{*Warning: ancient web site, with poor taste in animated gifs, and many broken links!}