This question is intimately connected with the first law of thermodynamics, aka conservation of energy. Kuhn did a detailed study of the question in his paper "Energy Conservation as an Example of Simultaneous Discovery" (reprinted in The Essential Tension). Also quite informative: the introduction by Mendoza to the collection Reflections on the Motive Power of Fire, and other Papers on the Second Law of Thermodynamics (reprint published by Dover books). As Mendoza says in his introduction:
The terms "heat" and "caloric" must be discussed in more detail. ... The most important fact which emerges [from studying papers and textbooks, in the period 1780-1836, in France] is that the caloric theory, which implied that heat was conserved in all thermal processes, and the theory that heat was equivalent to work were both regarded as true by the French physicists...
He backs this up with quotations from a joint work by Laplace and Lavoisier (1780) and a book by Lamé.
As Mendoza explains, one should not unequivocally identify caloric with heat:
In fact, there are two heat quantities that it is useful to define [i.e., heat energy and entropy]... Neither quantity is any more fundamental than the other. It is through a historical accident, an arbitrary choice, that we happen to call one of these quantities by the familiar term "heat" and the other by a pseudo-Greek name.
If you study Carnot's famous essay carefully, you will find that usually (not always), his assertions are accurate if you replace the term caloric with entropy. One place where this fails is in his discussion of the latent heat of water vapor; however, his conclusions agreed with (erroneous) experimental results by Delaroche and Bérard.
In the 1840s, James Joules' experiments increasingly convinced scientists that energy was conserved, or in other terms, that the mechanical equivalent to heat did not depend on the method of conversion. (This is discussed in great detail in the paper by Kuhn I cited above.) As a result, Thomson (later Lord Kelvin), Clausius, and others took a fresh look at thermodynamics. (The book by Crosbie Smith, The Science of Energy: A Cultural History of Energy Physics in Victorian Britain gives a thorough account.)
The paper by Clausius "On the Motive Power of Heat, and on the Laws which can be Deduced from it for the Theory of Heat" (included in the Dover collection) gives a concise treatment of the issues. He reviews work by Joules and Mayer, and also cites experiments by Regnault that corrected the erroneous results of Delaroche and Bérard. The conclusion seems inevitable that heat cannot be conserved. He then quotes an early paper by Thomson:
If we abandon this principle [conservation of heat], we meet with innumerable other difficulties ... and an entire reconstruction of the theory of heat from its foundation.
I believe that we should not be daunted by these difficulties, but rather should familiarize ourselves as much as possible with the consequences of the idea that heat is a motion, since it is only in this way that we can obtain the means wherewith to confirm or to disprove it.
The rest of the paper reworks thermodynamics as it then stood, to be consistent with the first law.
- In Carnot's famous paper, and in the immediate aftermath, heat energy and entropy were not clearly distinguished; caloric had aspects of each. (By the way, neither of the terms energy nor entropy were in use at the time.)
- As the first law emerged from experimental results and new theoretical considerations, scientists did not, at first, see how to reconcile it with the body of well-established work already developed from the second law (e.g., Clapeyron's equation).
- Within a relatively short time (a couple of decades at most) classical thermodynamics emerged as we know it today.
Caloric was no longer viable after that: neither heat energy nor entropy is conserved, and the hallmark of caloric was its conservation.