The heat comes from the kinetic energy of the items being struck together. If you take two large steel balls, and place a piece of paper between them, and then strike them together, the paper will scortch at the point of contact, leaving a burned hole.
If flint is struck by steel, such as a hammer, it heats up at the point of impact as it absorbs the energy from the blow. The incandescent flakes of flint that fly off are hot enough to start a fire, if the flamable material they land on does not have enough mass to significantly cool it, and if there is enough air around the material. This is why dry tinder is used, as it has both of those properties.
The "flints" in cigarette lighters contain alloys of cesium, which acts like a real piece of flint, but also burns in the air, making them better at starting a fire, especially if they come into contact with naptha vapors or butane.
While not an expert in non-sparking tools, my surmise is that they spread the impact through a larger mass, or absorb the impact and do not chip possibly because they are softer or good heat conductors. But a non-sparking tool could still cause sparks by causing two chipping materials to come into contact suddenly. A copper hammer can cause a spark if dropped into quartz gravel.