Great question!  Well, it really depends.  Extreme cold is definately very hard on the bindings in particular.  Different bindings are made from different materials -- steel, nylon, plastic, foam, rubber, etc...  Each of these materials have varying quality levels and design specifications and thus will perform differently under extremely cold conditions.

That being said, most modern snowboarding equipment should function well down to zero degrees.  Below that and the materials stiffen up and chance cracking.  Well-made and newer equipment will function to tempratures down to what a human being can stand!  Wow, nice stuff!

Additionally, extreme cold slows down the snow.  I went out at -25 degrees one time and had to do Black Diamond runs or steeper only due to how slow the snow was.  There are special waxes made specifically for cold weather that help a lot, but still not nearly as fast a glide as say 30 degree snow.

Remember as well to PROTECT YOUR BODY in extreme cold.  Neck gators, face masks, and chapstick can be life-savers.