Origin

Dammit Jim, I'm a Doctor not a X is a meme which originated from Star Trek: The Original Series . The phase was originally said by Dr Leonard McCoy but is also said by many other characters not just restricted to The Original Series

Use in show

The phase is said often in the show and is said in later Star Trek series. Not always would McCoy say the term and not always exactly like that. Special thanks to Memory-Alpha.org for the following quotes.
Note: The Following are all said by Leonard McCoy, It it said many more times by other chracters. For a full list click here

• "What am I, a doctor or a moon-shuttle conductor?" (TOS: "The Corbomite Maneuver")
• "My dear girl, I'm a doctor. When I peek, it is in the line of duty." (TOS: "Shore Leave")
• "I don't know, Jim. This is a big ship. I'm just a country doctor." (TOS: "The Alternative Factor")
• "Me, I'm a doctor. If I were an officer of the line…" (TOS: "A Taste of Armageddon")
• "What do you mean what sort of work? I'm a doctor." (TOS: "This Side of Paradise")
• "I'm a doctor, not a bricklayer." (TOS: "The Devil in the Dark")

…to which Kirk replies, "You're a healer, there's a patient. That's an order."

• "I'm a surgeon, not a psychiatrist." (TOS: "The City on the Edge of Forever")

In the episode "Court Martial", Doctor McCoy is called as an expert witness in space psychology. Psychology and Psychiatry are completely different, however.

• "I'm not a scientist or a physicist, Mr. Spock…" (TOS: "Metamorphosis")
• "Look, I'm a doctor, not an escalator." (TOS: "Friday's Child")
• "I'm a doctor, not a mechanic." (TOS: "The Doomsday Machine")
• "I'm a doctor, not an engineer." (TOS: "Mirror, Mirror")

…to which Montgomery Scott immediately replied, "Now, you're an engineer."

• "I'm not a magician, Spock, just an old country doctor." (TOS: "The Deadly Years")
• "I will not peddle flesh! I'm a physician." (TOS: "Return to Tomorrow")
• "I'm a doctor, not a coal miner." (TOS: "The Empath")
• "I'm not a mechanic, Spock…" (TOS: "The Empath")

Use in Meme

Not all uses of the term have a exact formula but in most cases it follows "Dammit Jim, I'm a Doctor not a X."

More examples

A Shirt containing a parody of the meme
A tattoo of Leonard McCoy
A news story appropriately titled – via WaybackMachine
A Facebook group

Google Insight

damn it jim james t kirk star trek enterprise dr leonard mccoy bones memory-alpha.org i'm a doctor not a x