Should outdoor deck gazebos or grills be grounded?
If lighting hits an ungrounded metal structure, it will usually branch out at the bottom of the structure, following multiple paths between the structure and the ground.  Some of those paths may be horizontal for a considerable distance. 
 It will burn, explode, and/or electrocute anything in the way.  Grounding a metal structure makes it safer.

However, surviving a direct lightning strike takes a lot more than just a single copper rod in the ground.  To do it right, you'll need a soil analysis and a network of grounding rods that are bonded together.  I think a better idea is to stay away from the gazebo (and the deck for that matter) when there's lightning in the area.