scr4 and brain wreck are right, but I’d like to add something.
The wavelengths of absorption and emission are two different principles. When emitting light, most objects follow what is called a “blackbody spectrum,” and it doesn’t change too much among common substances. Ie, there is a sort of bell-curve of frequencies, and the peak moves toward higher frequencies (aside from also rising in amplitude) the hotter something gets. Things slightly above room temperature glow in the infrared, but heating things further will make them glow red, then white, then blue. People used to judge the temperature in a furnace based on the color of the thing glowing inside it.
When absorbing, the nature of the material matters a bit more. An electron in an atom will only absorb a photon of an energy such that it gets bumped from one orbital to another. Of course, it will also emit in the frequency also, so the nature of the material alters emission as well. However, because when absorbing you’re dealing with materials that are cool and have their electrons in mostly the same state, while when emitting you’re dealing with hot materials with electrons all over the place (and the phenomenon of emission deals with precisely those electrons which aren’t in the ground state), it just so happens that the nature of common materials affects the way they absorbs light more than it does how they emit it.
You make an excellent example of water and microwaves. However, the very phenomenon of color is that of absorption. If something is red, it means it’s not absorbing that frequency. I used to have a camcorder that could see in infrared, and some plastics (especially those used with devices that have remote controls), are clear in the infrared.
Another thing, food is heated using infrared/red heat lamps mostly because infrared will penetrate it better and won’t just crisp the surface.
So yeah, the connection between infrared light and heat is mostly circumstantial. Of course, heat is molecules moving, and the statement “infrared light is heat” is completely misleading when taken at face value.