In the Timaeus, written in the fourth century BC, Plato has his speaker build the physical world out of geometry and hand four of the five regular solids to the four elements. The tetrahedron goes to fire.
The reasoning is given, and it is worth reading as reasoning rather than as decoration. Plato argues that the elements differ in mobility and in sharpness, that fire is the most mobile and the most cutting of them, and that among the solids the tetrahedron has the fewest faces, the sharpest angles and the least bulk. The most piercing element therefore receives the most piercing shape. He is not reporting an observation. He is arguing that like should answer to like.
Underneath the assignment is a second structure that usually gets dropped. Plato does not treat the solids as fundamental. He builds their faces from two elementary right triangles, the half-equilateral and the half-square, and it is the triangles that are basic. That has a consequence. Fire, air and water are all made from the half-equilateral, so they share a building block and can be taken apart and reassembled into one another. Earth is made from the half-square, and Plato is explicit that it cannot turn into any of the rest. The tetrahedron being fire is a detail of the scheme. The triangles are the theory.
None of this is a statement about matter, and Plato claimed no experimental support for it. It is a statement about what one philosopher in Athens thought a rational cosmos ought to look like. Nearly every later association of the tetrahedron with fire, will, energy and action descends from this passage, whether or not the writer knows it. Where a modern source presents that link as ancient universal knowledge about the element, what it is actually reporting is a few pages of a single dialogue.