I think you are the
Dragon or demon, locked in its pen,
The demon core was a spherical 6.2-kilogram (14 lb) subcritical mass of plutonium, 89 millimetres (3.5 in) in diameter, manufactured during World War II by the United States nuclear weapon development effort, the Manhattan Project, as a fissile core for an early atomic bomb.
But without even moving, I once killed two men.
Harry Daghlian and Louis Slotin
Most things like me kill when they're open, exposed,
I think this is in reference to atomic weapons in general which are intended to kill outside of the lab environment.
Yet my murder made martyrs because I was closed.
The two criticality incidents are quite famous because they demonstrate how dangerous the demon core was, even in a research environment. As clarified by the OP, the reference to open/closed also hints at the second incident where a separation needed to be maintained between the two halves of the neutron reflector.
The first man I killed tried to pen me for sport,
"Dropped it," would go in the incident report.
On Aug 21, 1945, during a critical mass experiment in which the core was placed within a stack of neutron-reflective tungsten carbide bricks, Harry Daghlian accidentally dropped a tungsten carbide brick onto a plutonium bomb core. The mishap caused a critical reaction, and Daghlian quickly tried to knock the brick away, unsuccessfully, and resorted to removing the bricks by hand to halt the reaction. He stopped the reaction, but was exposed to massive amounts of radiation.
The second man I killed liked to play with my tail,
But a slip of the wrist and the warnings prevailed.
On 21 May 1946, with seven colleagues watching, Slotin performed an experiment that involved the creation of one of the first steps of a fission reaction by placing two half-spheres of beryllium (a neutron reflector) around a 3.5-inch-diameter (89 mm) plutonium core. Slotin grasped the upper 228.6 mm (9-inch) beryllium hemisphere with his left hand through a thumb hole at the top while he maintained the separation of the half-spheres using the blade of a screwdriver with his right hand. This manoeuvre was often referred to as tickling the dragon's tail. The screwdriver slipped and the upper beryllium hemisphere fell, causing a "prompt critical" reaction and a burst of hard radiation. Slotin subsequently died from the dose.
A killer they called me, and killed me for wrath.
Yet many before and after have walked the same path.
The creation of a critical mass was not unusual at that time. The core was later melted down and the material recycled for use in other cores.