The above glib, simplistic reply to the question as you posed in the title is, in fact, accurate. I replied that way to just make the point.
STDs are rarely if ever transmitted by saliva, which inhibits or kills many pathogenic bacteria and viruses; and the amount of genital secretions that can be transmitted by such indirect contact is generally to little for transmission to occur. Further, the skin of the penis, like the skin of your arm or hand, is resistant to infection unless there is an overt wound.
As for your symptoms, there are no STDs that would cause the symptoms you describe, unless there were simultaneous discharge of pus or mucus, sores on the outside, etc. Your symptoms are, however, quite typical for genitally focused anxiety, which I suspect is the main issue here. And as you suggest yourself, a physical or chemical irritation also is possible; a sexual lubricant getting into the urethra could cause it, as could minor trauma from vigorous massaging.
A few days ago I replied to a question that offered an opportunity to address the biological reasons STDs are rarely transmitted by exposures other than penile insertive intercourse. The link to that thread is below, as is an older one that makes many of the same points. (In the second link, read the entire thread; the important points are in the follow-up comments.)
http://www.medhelp.org/posts/show/1861534
http://www.medhelp.org/posts/show/1119533
If your symptoms persist, or if you develop more typical STD symptoms (blisters/sores of the penis, discharge of pus or mucus), of course see a doctor or clinic. However, based on the exposure and your symptoms, I really don’t think you have anything to worry about. Assuming things clear up on their own, as I think they will, I don’t recommend STD testing and believe it is safe for you to continue unprotected sex with your regular partner if you have one.
I hope this has helped. Best wishes-- HHH, MD