Why do we feel heat from infrared light but not from shorter wavelengths?

The short answer is: of course we do.

The long answer has a few parts.

Absorption

Different wavelengths have different absorption ratios in the same materials. The typical example is a plastic bag, which is transparent to visible light, but opaque to infrared light. This means that it mostly lets visible light through (no absorption, no heating), while capturing infrared light (absorption, heating).

The human body is mostly transparent to both very high and very low frequency light. Radio passes straight through, and so do e.g. X-rays for the most part (don't try hiding from a nuclear blast behind another human - not a lot of protection). There could be kilowatts of radio waves passing right through your body without you noticing any heating, because your body only absorbs very little of those frequencies. Infrared is very important because it's readily absorbed in water - and there's a lot of water in a human body. Still, visible light is readily absorbed in the human body as well - you do in fact feel the heat of visible light (if you've ever tried focusing a lens on a piece of paper - you're mainly doing this with visible light; infrared light of course has a different focus). However, under normal conditions, this tends to be dwarfed by infrared light, because...

Emission

... most of the light sources around us are pretty close to black body emitters. You may be familiar with the rather distinctive curve derived from Planck's Law for the photon emission of a black body. Now compare the area under the curve in the IR region with the one in the visible or UV region - for low-temperature sources (simple incandescent lightbulbs) IR utterly dominates, and even for sunlight, you can see that even before accounting for all the trickiness of the atmosphere etc., we get a whole lot more IR light than visible light. While the per-photon energy of UV light is much higher than for visible light, the total amount of energy carried by all the photons is much smaller - and most of UV light is absorbed in the atmosphere anyway.

In fact, even modern high-efficiency light bulbs still tend to produce more IR light than visible light; light sources with efficiency higher than 50% are quite rare. A decent LED light bulb might have an efficiency around 20%, which means that for each watt of light, it emits four watts of heat (either direct IR radiation or cascading through its surroundings).

IR is everywhere

The feeling of heat on your skin is a relatively simple matter of comparing two temperatures - the temperature of upper skin, with the temperature of lower skin. If the upper skin is hotter, we feel warm, if it's colder, we feel cold.

All objects emit IR light. All of them - and in proportion to their temperature. That's why IR is commonly associated with heat - the room around you is hot with IR radiation, the computer under your desk is hot with IR radiation, you are hot with IR radiation. That's what makes passive thermal vision work - different objects have different temperatures and different emissivity, which makes them stand out against each other on an IR sensor.

How much heat are we talking about? Let's compare to the Sun, just for fun. Sunlight gives about 1100 W per square meter on ground level (there's plenty of different averages - this is basically the value at noon on the equator with average cloud cover). Out of this, about 55% is infrared light and about 42% is visible (see? Even after all so much IR is absorbed in the atmosphere, it still dominates on ground-level :)). So let's say you get about 500 W of IR heat on surface level per square meter. Not something to sneeze at, certainly. Let's put it in human terms, though.

Take a naked human and angle him to the Sun. The human surface area on average is about two square meters, and one half of that is facing away from the Sun, so on a great day, you might absorb as much as that 500 W of IR light. Close enough for our purposes :) But you have to consider something else - the human body is also an IR emitter, and quite a good one at that. How much energy does a typical human emit when idle? About 1000 W. Yes - almost the entire incoming sunlight in the most sunlit place on Earth at noon. So why do we feel warm anyway?

Because sunlight is not the only source of radiation on Earth. Humans radiate a huge amount of energy, true - but so do our surroundings. If you close yourself in a dark room at room temperature, you'll get about 900 W back. So your net radiative loss is only 100 W, rather than 1000 W. And it so happens that the average idle heat loss of the human body is around 100 W, which is why a 25° C room with no direct sunlight feels comfortable - it's more or less a perfect balance between the inefficiencies of human metabolism and the difference of temperature between the human body and the room. Of course, this changes a lot depending on clothing and other factors. Add a 100 W light bulb, and you're outright warm :)

IR photons have a very low energy

Now, this might sound counter-intuitive, and that's because this mostly targets human thinking, rather than reality. But for completeness: infra-red light has negligible effects beyond heating. It's not energetic enough to affect atoms or chemical bonds. The only thing it maps well to is the random motions of atoms and molecules - which add up to what we call heat.

On the other hand, if you take something like visible light, beyond the vibrations you also get chemical changes - electrons being bumped into excited states, (relatively weak) chemical bonds changing; in fact, that's why we see visible light in particular - it's more or less in the sweet spot of "strong enough to excite electrons, but weak enough not to destroy the photoreceptors and their proteins" (the animals that are sensitive to IR use a different mechanism than electro-chemistry). UV light can be easily absorbed, but it's strong enough to break even rather strong chemical bonds, which leads to substantial damage - that's how UV light destroys the DNA in your cells, for example (though again, there are animals that have UV senses - many insects do).

So there's this weird bias in the human mind - you see all those different kinds of light, and they all have interesting properties... except for IR. It just heats stuff, and not much more. Go to even deeper IR (like microwave or radio waves), and you get other interesting behaviours - and a lot less direct heating, since they are less easily absorbed.

Conclusion

We mostly care about infra-red radiation in terms of heat, simply because there's so much of it everywhere, and most of the sources of visible light also involve a higher amount of infrared light. However, take a pure visible light source of enough wattage (say, a cold, high power LED bulb) and point it at yourself, and you'll feel the heat. We use a lot of high-powered visible light lasers, and they're quite obviously pretty good at heating things.

A typical photo-voltaic solar panel captures most of its power production from visible light, as do photosynthetic plants (while some plants also need UV light, that's really more of an catalyst, rather than the primary source of energy; consider how well your house plant is doing despite getting no UV light at all). You need an energy gradient to do useful work, and that makes visible light a lot more interesting than IR for most plants - take a look at an IR photograph of trees or plants; there's quite a decent chance their leaves are actually reflecting incident IR light rather than absorbing it, simply because it's basically waste heat you do not want. That said, there are photosynthetic organisms that are different - working with IR, red or blue light, depending on their niche.

This is probably due to Planck's radiation law and Wien's displacement law giving the wavelength of maximum energy emission, which shows that for temperatures of usual very hot bodies on the order of (a couple) $1000K$, the radiation energy emitted in the infrared/visible ($\lambda>380 nm$)region is much larger than in the ultraviolet region ($\lambda<380 nm$).

We do. Here are two ways to demonstrate it.

First the recommended way: Get a really bright white LED (e.g. a 1200 lumen bike light) and look at the spectrum either on a data sheet or witha spectrometer. If you don't trust that, put some IR-blocking glass in front (e.g. KG1). Put your hand in the beam. You'll feel some warnth especially outside on a cold night. A variation is to get an extremely bright single-colour visible LED.

Now the not-recommended way: Put your hand in the beam of a visible laser of at least 50mW (more if the beam is wide). 120mW of 532nm (green) into a sub-millimetre spot on the back of your hand gives quite a sting. With this sort of power your should be wearing goggles, but then you can't see the beam and can get your hand in it accidentally when aligning. But don't try this at home.