How dangerous is diplopia?

Sometimes. If it is monocular (you cover one eye and the double vision is still present), not worrisome. If binocular (you cover either eye and it disappears) then you should see an ophthalmologist.
Sometimes. Depends upon the cause. If it is due to injury the danger is that you will be seeing double and can have navigation and driving issues. If it is due to tumor, vascular malfunction or trauma, then more serious accompaniments may be present that need medical treatment so in these situations it could be dangerous.

Related Questions

How dangerous is diplopia while driving?

Serious problem. Diplopia, the seeing of two images at the same time due to misalignment of the eyes, is a very disturbing pattern for someone who acquires this. When driving, you cannot tell which is the real image and which is the secondary image. You should avoid driving, or temporarily cover one eye. Hopefully you are driving to the neuro-ophthalmologist to evaluate your diplopia. Read more...