I love tan!

Rewrite the equation as:

$\frac{2\tan\theta}{1-\tan^2\theta}=3\tan\theta$

Note that the left side is undefined for $\theta=45^\circ,135^\circ,225^\circ,315^\circ$ while the right side is undefined for $\theta=90^\circ,270^\circ$ . In particular, we can thus multiply both sides of the equation by $1-\tan^2\theta$ , since for all such other values of $0^\circ\leq\theta \leq360^\circ$ , $1-\tan^2\theta \neq 0$ . We then have:

$2\tan\theta=3\tan\theta(1-\tan^2\theta) \Rightarrow \tan\theta(3\tan^2\theta-1)=0$

For $\tan\theta=0$ , we have $\theta=0^\circ,180^\circ,360^\circ$

For $\tan\theta=\sqrt{\frac{1}{3}}$ , we have $\theta=30^\circ,210^\circ$

For $\tan\theta=-\sqrt{\frac{1}{3}}$ , we have $\theta=150^\circ,330^\circ$

The sum of all these values is then thus $\boxed{1260^\circ}$