A cursory search finds no supporting (non-anecdotal) evidence for or against smoking oregano for recreational usage.
A 1972 article from Human Pathology mentions oregano as a substance that can be passed off as pot to fool the buyer:
Occasionally the buyer is "burned" and finds that he may be smoking oregano, catnip, or even grass clippings.1
The US Food and Drug administration does not regulate oregano beyond checking it for insects, filth, or mold.2
Finally, there is a bunch of stuff that oregano might do medicinally, but it seems like it would relate mostly to its anti-fungal or anti-bacterial properties. A list of articles on that topic can be found at GreenMedInfo, a site that "provide convenient and open access to the biomedical research available today on the therapeutic value of natural substances in disease prevention and treatment."3
Please note that this answer does not constitute medical advice. It is only meant to summarize published research related to the topic and limited the cited sources. Consult your physician about what these results may mean for your health.