Even though you may have a filled octet, if the central atom has a positive formal charge, we generally form double bonds until the formal charge is reduced to 0 as close as possible as that will be the more stable configuration. However, there are some exceptions.
Please take a look at the chlorate ion. For chlorate, you would expect to leave four single bonded oxygens to the chlorine, however, we are left with a formal charge of +3 on the chlorine and -1 on each of the oxygens. Therefore, we form double bonds until the formal charge is removed and are left with only a formal charge of -1 on the single bonded oxygen.
edit: looking at your question the structure on the left is preferred even though the one on the right has a lower formal charge.
Perhaps Wikipedia has the wrong structure for the Nitric acid page? It looks like all the oxygen atoms are single bonded in the figure to the top right.
https://en.wikipedia.org/wiki/Nitric_acid
Looking for images, Wikipedia does have the right resonance structures with the double bonded oxygen, however, there is only one double bond contrary to what we expect which is two double bonds.
https://commons.wikimedia.org/wiki/File:Nitric-acid-resonance-A.png