Has anyone conducted experiments demonstrating the conversation of energy or is it simply assumed to be true? Testing the conversation of energy would entail accurately measuring the joules of the energy input and output, including any energy lost to friction or heat. That would mean not simply assuming "some" energy was lost, but actually measuring the energy that was lost.
The law of energy conservation has always been verified experimentally at the cost of often difficult measurements. In a way, Joule's experiments (mechanical equivalent of heat) are a good example. Another interesting example is the neutrino hypothesis by Pauli :
In 1930, this evidence was problematic for physicists working in the field. What happened to the law of conservation of energy for beta decay? The seemingly missing energy even led Niels Bohr to propose doing away with that most fundamental conservation law. A mortal sin for a physicist.
Every experiment has shown us that conservation of energy holds true. This is not only the case for macroscopic objects, but also for interactions at the quantum level$^1$. Many physicists at different times in history confirmed this conservation law, most notably James Joule and Nicolas Sadi Carnot.
In cases where energy is lost to friction or heat, although it is a little harder to measure, we still find that the total energy of the system is constant. We can write this law mathematically as $$U=U_i+W+Q$$ where $U$ is the total energy of the system, $U_i$ is the initial energy, $Q$ is the heat added or removed from the system, and $W$ is the work done on or by the system.
Conservation of energy is a fundamental law of nature, and there has not been an instance where this is law is not observed (there is debate about this$^2$ in cases such as universal expansion, gravitationally redshifted photons, some unusual circumstances in cosmology and in certain applications of general relativity).
Noether’s theorem tells us that certain conservation laws result from symmetries. The law of conservation of energy results from time translation symmetry. So it is applicable to systems that have this symmetry. For almost all physical systems, this symmetry, and therefore the law of conservation of energy holds true.
$^1$ It was once thought that energy was not conserved for beta decay since at that point in time, when nuclear beta decay was studied, it was found that the decay products and the energies were not consistent with conservation of energy. Some other mechanism had to be at play such that energy was conserved, otherwise physicists would have to accept that energy was not always conserved in fundamental processes. This idea was extremely unpalatable, and so it was hypothesized that there must be an additional decay particle, that was carrying away energy such that the total energy was conserved. This prediction was made by the physicist Pauli in 1933, and he called this particle a "neutrino" and in 1956, 23 years later, the neutrino was detected experimentally. The reason why it was hard to detect it originally, was because it had no charge and was virtually massless. This once again cemented the idea that energy is a conserved quantity.
$^2$ The total energy of an isolated system is always constant. If we consider the universe to be an isolated system, one can say that the total energy in the universe is conserved.