What is the accurate model of infrared light vs. heat?
The exact details depend on the chemical composition of the material, but for a simple model, check out https://en.wikipedia.org/wiki/Black-body_radiation
All physical objects emit electromagnetic radiation corresponding to their temperature (as opposed to reflected, reaction-induced, or stimulated radiation, more below), with a corresponding blackbody radiation curve. Note that blackbody emissions don't have a specific frequency (say, unlike laser light), though the curve has a peak emission.
Traditional incandescent light bulbs are blackbody emitters, as is a hot coal, the Sun and other stars, or the glowing elements of an electric radiant heater. Human eyes see blackbody radiation, in the visible range, as having a characteristic colour, which paradoxically gets bluer as the temperature gets higher, so the ruddy tones of low-voltage incandescent lamps are low colour temperature, and the intense white of halogen lamps are high colour temperatures. Blackbody radiation and colour temperatures are given in Kelvin, with typical visible light corresponding roughly to ~3,000 - 8,000K. You might also recognise these values from gamma or colour correction values on monitors and video equipment.
(Colour temperatures were used to judge processing for ceramics and metal processing / firing / smelting processes as well. They're also used to classify stellar temperatures, and in conjunction with brightness and/or distance estimates can be used to find stellar sizes.)
Infrared light is characteristic of peak blackbody emissions of bodies at or near "room temperature". So IR is not heat, but is associated with hot objects. Heat itself is ... thermal energy, which is its own complex thing.
Since all EMR carries energy, all EMR heats objects in which it is absorbed. A sufficiently intense emission above the IR range will also heat an object. However hot objects in near proximity transfer a great deal of thermal energy directly as IR emissions. The illuminating capability of "white" light (typical solar emissions) allows a small amout of EMR energy to provide a high degree of visual information without imparting much heat on the illuminated objects (though it does impart some). Consider that even an (inefficient) 100W incandescent bulb, whilst hot to the touch, does not significantly heat the objects it is illuminating, and a far more efficient equivalent LED lamp, drawing about 15W of electrical power, accomplishes the same illuminating power with vastly less heat. We don't have to toast things to be able to see them, just bounce a small amount of (high-energy) visible light off of them.
Not all light (or more accurately, EMR) is blackbody radiation. There can be chemically-emitted light, directly from chemical reactions. The blue glow of methane (natural gas) combustion is a chemical emission, contrasted with the yellow-white glow of a candle, actually blackbody emissions of suspended soot particles in the smoke plume. (There's a blue region of chemical emission at the base of the flame.)
Chemical emissions are based on specific frequencies of EMR emitted as electrons transition between energy states, if I understand correctly are a quantum phenomenon. Fluourescent and LED lamps also work based on chemical / valance emissions, and operate in far narrower bands than blackbody emissions -- one of the reasons these lamps can appear harsher than incandescents. Similarly various chemical lamps, especially high- or low-pressure sodium vapour, formerly popular as street lighting, which emit in narrow bands (low-pressure especially).
Some forms of ionizing radiation are also EMR emissions, at far higher energy levels, triggered by nuclear rather than electron emissions. Typically gamma-rays.
And stimulated emissions such as lasers and masers are ... another phenomenon I understand only poorly, but are also tuned to very tight frequencies. Radio and microwave emitters are somewhat similar.
https://en.wikipedia.org/wiki/Black-body_radiation