Is azide paramagnetic? Is the azide ion ($\ce{N3-}$) paramagnetic? 

How do I use molecular orbital theory to show that it is or isn't?

I drew a sine wave diagram and filled up the $1\pi$ and $2\pi$ MOs, and since there are no unpaired electrons, can I conclude that the ion is not paramagnetic?
Deriving a proper MO digram for N$_3^-$ is not completely trivial a task (e.g. a random google hit, which seems to me well done: www.chem.uci.edu/~lawm/Azide%20MO%20Diagram.pdf). In the end you will have to show that the degeneracies of the orbitals together with the electron count will not require open shells. Beware of Lewis-structures and mesomery, think of O$_2$, it looks perfectly o.k. to draw O=O, only the MOs and their degeneracies will reveal that its paramagnetic.