A triple-bond is not inherently unstable. In chlorine azide, though, you've got a strongly electronegative halide (chlorine) bound via tenuous single-bond to a partial triple-bond in the nitrogen compound. One of those nitrogens has a partial charge at all times.

Bottom line: that chlorine wants to dissociate, badly. When it does, it's going to take the azide with it. It's like a chemical hair-trigger, waiting to be pulled by the nearest partial charge.