In science, material theory of heat, aka "substance theory of heat" (Planck, 1949), “matter theory of heat”, or “fluid theory of heat”, give or take, is a heat theory which supposes that heat is a type of matter or particle, with or without mass; it was eventually superseded by the “kinetic theory” (Hermann, 1716) and or “kinetic theory of heat”, which was itself later subsumed into the “mechanical theory of heat” (Clausius, 1865).

Overview

The ancient roots of the theory that heat is a type of matter originated, in a general sense, from the three element theory, four element theory, and five element theories of the Greeks, which supposed that fire was one of the elements. This logic, however, began to loose sway in the 18th and early 19th century; the following is one take on this:

“Nothing ever delighted me so much as the discovery that there were no elements of earth, fire, or water.”
— Percy Shelley (1804), on his age 14 education [2]

In short, in respect to the nature of heat, as explained by the “four element theory” (Aristotle, c.330BC), became fire explained by the “three principles theory” (Geber, 790), which became heat explained via the “sulfur combustion model” (Paracelsus, 1524), which became the “terra pinguis model” (Becher, 1699), which became the “phlogiston model” (Stahl, 1703), which became the “caloric model” (Lavoisier, 1789), which, following the disproof of the caloric theory, by Benjamin Thompson (cannon boring experiment, 1798) and Humphry Davy (ice rubbing experiment, 1799), the kinetic theory of heat and later mechanical theory of heat usurped all the former models.

Quotes
The following are related quotes:

“Despite the caution of Black and his followers, in the second half of the eighteenth century, there was a hardening of opinion in favor of the ‘material theory of heat’. The difficulties of the ‘kinetic theory’ which was favored by mathematicians and physicists were still unresolved, while the chemists who came increasingly to dominate the study were, perhaps by tradition (in the footsteps of Boerhaave), perhaps by metaphysical preference, predisposed to favor the substantial theory. An exception was Pierre Macquer, but he had few disciples.”
— Donald Cardwell (1971), From Watt to Clausius (pg. 57)

“Blagden, in his obituary on Cavendish (Mar 1810), failed to mention Cavendish's work on heat, although he made note of all of his other major work. Odd, I say, because in none of his other work was Cavendish known to have declared himself publicly a more decided follower of Newton, according to his understanding of Newton, than in his work on heat, and also because Blagden assisted Cavendish in this work. If there is a circumstance that might bear on Blagden's neglect or forgetfulness in his obituary of his late friend and colleague, it is that Blagden subscribed to the popular ‘material theory of heat’, of which Cavendish held a low, almost contemptuous, opinion.”
— Russell McCormmach (1988), “Henry Cavendish on the Theory of Heat” [1]

“In 1786, in a book on the latest advances in heat, light, and pneumatic chemistry, Bryan Higgins wrote that there was no need for him to justify his ‘material view of heat’, since Cavendish together with "other distinguished philosophers have accepted it." If Cavendish took note of Higgins's mistake, he must at the same time have realized that he had not made sufficiently public his view of heat. With ‘heat’, Cavendish intended to set the record straight.”
— Christa Jungnickel (1999), Cavendish: the Experimental Life (co-author: Russell McCormmach) (pg. 415)

See also
● Wave theory of heat (Ѻ)

References
1. McCormmach, Russell. (1988). “Henry Cavendish on the Theory of Heat” (abs), Isis, 79(1): 37-67.
2. Rossetti, William R. (1866). A Memoir of Shelley: with Fresh Preface (pg. 8). Shelley Society.

Further reading
● Freudenthal, Gad. (1995). Aristotle’s Theory of Material Substance: Heat and Pneuma, Form and Soul (Amz). Clarendon Press.