Why can some electromagnetic waves heat things up while others cannot?

In a solid, "heat" consists of random vibrations of the atoms in that solid around their equilibrium positions. If the radiation striking that solid has a wavelength component that is close to one of those possible vibration modes, then the radiation will couple strongly with that vibratory mode and the solid will accept energy from the incident radiation and its temperature will rise.

If the incident radiation has too high a frequency (X-ray or gamma) the coupling is poor and the radiation just goes right through without interacting much. If the frequency is too low (radio frequencies lower than radar) the radiation bounces off and also doesn't interact much. This leaves certain specific frequency bands (like infrared and visible light wavelengths) where the interaction is strong.

Note that this picture is somewhat simplified in that there are frequency bands in the gigahertz range where the RF energy bounces off electrically conductive materials like metal (this gives us radar) but interacts strongly with dielectrics and materials containing water molecules (this gives us microwave ovens).

Note also as pointed out below by Frederic, molecules possess resonant modes that their constituent atoms do not and these can be excited by RF energy as well. Many of these molecular modes lie within the infrared range, giving rise to the field of IR spectroscopy.

As mentioned by niels nielsen, the EM waves get absorbed when their wavelength matches the vibrational modes of the atoms in the solid. This causes the atoms to vibrate even harder and thus rising the temperature. From the vibration of atoms in solids, the extension can be made towards the vibration of polymers and organic molecules, which have additional vibrational and rotational modes. For example, in organic molecules, the entire molecule can vibrate (in addition to the single atoms in it). This vibration happens on a different length scale and therefore EM radiation with different (lower) frequencies can get absorbed with respect to regular solid materials. Furthermore, also rotational modes exist in these organic molecules which can also absorb EM radiation and give rise to heating.

As humans are made of these organic molecules, it are these molecular vibrational modes that absorb the IR radiation and that give us the feeling of temperature.

ALL electromagnetic waves transfer energy. When they meet some body, they are either absorbed, scattered, or partially both.

The part that is absorbed always heats the absorbing body. Absorbed EM waves may, or may not do other things as well (like chemical changes or electric currents).

The reason why you don't feel much heat from a LED bulb is because the LED bulb doesn't radiate much. An oven is 1-5kW, the Sun is some 1 kW / square meter at noon. A typical LED bulb is 3-15W.

You can try some powerful LED (there are e.g. 50W directed LED headlights that can burn your skin pretty much).

It is very important to understand that heat energy is stored in the degrees of freedom of molecules.

Heat energy, at a microscopic level, is stored in the degrees of freedom of atoms and molecules. These degrees of freedom are translational, rotational and vibrational. They all store different amounts of energy, depending on the geometry of the atom. Translational degrees of freedom are the atom or molecule moving around in space, and there are always 3 for the 3 dimensions of space. The rotational and vibrational modes come from the geometry of the atom/molecule.

How is heat represented on a quantum level?

Now there are mainly three types:

1. translational

Translational degrees of freedom arise from a gas molecule's ability to move freely in space.

2. rotational

A molecule's rotational degrees of freedom represent the number of unique ways the molecule may rotate in space about its center of mass which a change in the molecule's orientation.

3. vibrational

The number of vibrational degrees of freedom (or vibrational modes) of a molecule is determined by examining the number of unique ways the atoms within the molecule may move relative to one another, such as in bond stretches or bends.

https://en.wikibooks.org/wiki/Statistical_Thermodynamics_and_Rate_Theories/Degrees_of_freedom

Now you are asking, why do certain wavelength photons heat up certain materials' molecules only while others cannot?

Every molecule has its own quantum mechanical characteristics, which includes the translational, vibrational and rotational modes' characteristics, and what wavelength photons those can correspond to. This means that certain wavelength photons energy needs to match the energy gap between those modes.

If the energy of the photon matches (or sometimes is exceeding) the gap between two modes, then the photon might be absorbed with high probability.

Now it is not just that simple. Certain wavelength photons do have the ability to transfer their energies with higher probability to molecules that have a certain type of available degrees of freedom (mode).

Thus, certain molecules that have degrees of freedom available in the different translational, vibrational or rotational modes, can be excited by different wavelength photons.

Just a note, the other answers do not address this, but heating a material is contrary to popular belief not only mainly by absorption. A lot of photons' energy is transferred by inelastic scattering. In this case, the photon does not cease to exist, and only transfers part of its energy to the molecule.

https://en.wikipedia.org/wiki/Inelastic_scattering

Things heat up when they absorb radiation. They do not heat up if they are transparent to that radiation or if they reflect that radiation. When they are transparent, the radiation passes through without losing much energy.

Different wavelengths have different absorption profiles in different materials (owing to the materials' atomic structure) as mentioned by @nielsnielsen and @Frederic. Glass looks pretty transparent in the visible spectrum, but absorbs infrared and ultraviolet radiation, so it's opaque for these wavelengths. I used to operate an infrared spectrometer. Instead of using glass to mount the samples, we had to use disks made of salt, since salt remains transparent in the infrared range. Water also appears to humans quite transparent, but strongly absorbs microwaves (the basis for microwave ovens).

What I haven't seen explicitly in the other answers is that the radiation is absorbed at the atomic and molecular level when the photon energy ($E = h nu = h c/lambda_mathrm{vacuum}$) is equal to the energy required for quantum transition between different modes. These modes can include electron transitions in atoms, transitions of delocalized electrons in molecules, vibrational transitions of atomic nuclei, rotational transitions of atomic nuclei, and displacements of atoms in crystals.