The Shriners organization is called the Ancient Arabic Order of the Nobles of the Mystic Shrine, or A.A.O.N.M.S. It was established in New York City in 1870, and is an appendant body to Freemasonry. From its inception, the focus was on camaraderie, and Shriners use the phrase "Pleasure without intemperance, hospitality without rudeness and jollity without coarseness." The Shriners are a fraternal organization, and its members are bound by oaths of loyalty. The membership is limited to men that are third degree, or master Masons. Apparently there is no direct connection to Islam, even though they have a Middle Eastern theme in their dress and logos.
Freemasonry is shrouded in mystery, and many consider it to be a quasi-religious body. Conservative, fundamental churches commonly do not allow oath-bound members of such organizations to be members of their churches. Externally, Freemasonry exists as a social/fraternal organization and seems to exist primarily for social purposes. The belief system of the Shriners arises from their association with Freemasonry, but they do not seem to aggressively recruit members or actively propagate the teachings of the parent body.
The Shriners have several socially beneficial projects. The most significant one is the Shriners Hospitals, which they control through the Imperial Council, their governing board. While the Shriners do some good and admirable things, due to their connection with Freemasonry, Christians should not be members of, or associated with, the Shriners.
For more information, we strongly recommend Ex-Masons for Jesus.