Is oxygen an organic or inorganic molecule?
Neither! By itself it is an element, or just an atom.
Oxygen is neither an organic nor inorganic molecule, because by itself it is just an atom. A molecule is two or more atoms joined together.
Then again, if two oxygen atoms bond, this would form an oxygen molecule, however the classification of this is a little tricky. It is definitely not organic, as this would require it to contain carbon, and most likely, hydrogen. Oxygen is usually classed as one of the components of an organic molecule, instead of being one on it's own (it could also be part of an inorganic one though, like a metal oxide).
However, it isn't really inorganic either. Inorganic molecules are more often known as inorganic compounds, which clears up cases like this.
I hope this helps, let me know if you need anything clearing up or expanded on. I should get back to you in a day or two.
Here are some references,
http://www.chemistryland.com/ElementarySchool/BuildingBlocks/BuildingOrganic.htm (not the best, but suitable for a bit of extra cross referencing)