Are there unabsorbable wavelengths (frequencies) in the electromagnetic spectrum? The electromagnetic spectrum is continuous. It can have any frequency since the ways of emission depend on velocity of electrons, energy levels, or vibrational modes of connections in molecules. On the other hand, the absorption of wavelengths depends only on vibrational mode of the absorbing material or energy levels of electrons in its atom. If vibrational modes and energy levels are both discrete and the emission spectrum is continuous, then doesn't it mean in practice that in principal there are unabsorbable frequencies or holes in the absorption spectrum?
 A: Why restrict consideration to atomic or molecular material? Once you have charged particles then absorption at any frequency is possible via the processes of photoelectric absorption, inverse bremsstrahlung, Compton scattering to name a few "continuum processes". These are known as free-bound or free-free processes and can produce absorption at essentially all wavelengths.
You seem particularly interested in the continuum absorption properties in the infrared. Here, for example it is well-known that water vapour absorbs both at discrete frequencies, associated with rotational and vibrational models of the water molecules, and over the entire infrared continuum (for example see these lecture notes. My understanding is that this continuum absorption is caused by a combination of thermal broadening of the many molecular transitions as well as collisional or pressure broadening that depend on the density of the absorber.
In the solid state then absorption bands that absorb light at a continuum wavelengths become possible. This occurs because of long range correlations and lattice vibrations. When atoms and molecules are not isolated then their energy levels are not discrete and are instead represented by a density of states function. Transitions between energy states in this continuum of course produce continuous absorption with wavelength.
A: We clever humans have found continuous absorption detectors from the very long radio wavelengths (a long wire) to the short radio and mm wavelengths (capacitors), to the ir/visible/uv (bolometers, CCDs and photodetectors), to the x-ray (gas detectors, bolometers, germanium CCDS, etc) and gamma rays (scintillation detectors).   These detectors have had their frequency dependent efficiencies measured in great detail and there are no holes.  I suppose at either end of the spectrum their could be technological problems (a radio wave with more than a thousand mile wavelength would be hard to detect I suppose, and maybe there is a frequency that is too high to detect.)
CCDs are solid state detectors that make use of a wide bandgap in which electrons or holes can be freed from their bond and be carried away.  The width of the bandgap determines the width of the frequency response.  A bolometer is just a metal that heats up when the photon gets absorbed.  The conduction band and the valence electrons in the solid metal absorbs over a wide range of frequencies.  In the radio band, one is also using the electron in the conduction band of a metal to respond to EM waves over a broad range of wavelengths.  Photometers use the fact that as long as the photon has sufficient energy, it can free electrons from the surface of a metal.
A: One aspect hasn't been covered by the other answers: If the absorber has an non-zero velocity wrt. the emitter, then in its reference frame the photon will be red- or blueshifted. Since any velocity (less than $c$) is possible, an arbitrarily high red-/blueshift is possible. Thus, any photon can be absorbed by any absorption process; it is just a matter of relative velocity (at least in principle; of course to absorb a radio wave with an X-ray transition, you'd have to move extremely fast).
Resonance scattering and Doppler shift
A less-drastic example of this mechanism is the "resonant scattering" of Lyman $\alpha$ photons, i.e. photons with an energy corresponding to energy difference between the ground state and the first excited state of neutral hydrogen: Only if the wavelength of a photon is very close to 1215.67 Å will it be absorbed. If it's just 1 Å lower (so that $\lambda\simeq1214.67$ Å and the photon is "bluer"), the probability of absorption is six orders of magnitude smaller. But if the atom has velocity component in the direction of the photon's path of 250 km/s away from the photon (which is entirely possible e.g. in the case of outflowing gas from a galaxy), then in the reference of the atom the photon's wavelength is roughly the 1215.67 Å needed for the absorption being highly probable.
(the photon will then excite the hydrogen atom which will decay after $10^{-8}$ seconds, emitting another Lyman $\alpha$ photon with $\lambda=1215.67$ Å in the atom's frame. If the new direction happens to be roughly back where it came from, then in the "global" reference frame, it will be redshifted, effectively converting a $\sim$1214.67 Å photon to a $\sim$1216.67 Å.)
Cosmological redshift
Lyman $\alpha$ lies in the ultraviolet. If it's emitted from a distant galaxy, then it is continuously redshifted on its way toward us, and we may observe it in the infrared (which is fortunate, because UV tends to be blocked by the atmosphere).
Thermal broadening of the absorption line profile
From your comments, I gather that you have some IR light incident on some molecules. If the molecules is a gas, then their velocities will have a ("Maxwellian") distribution, so that even if the energy of a photon doesn't match exactly the energy of a transition, there will probably be an atom that is moving with the right velocity to redshift the photon into resonance. The effect is that the absorption line profile is "thermally broadened", increasing the probability of absorption for off-resonance photons.
If the velocity distribution of the atoms is Maxwellian, then in one direction (e.g. along the path of the photon) it will be Gaussian. The natural line profile of the transition is Lorentzian, and the resulting line profile is then the convolution of the two. This is called a Voigt profile, and is dominated by the Gaussian in the line center, but by the Lorentzian in the line wings.
Pressure/collisional broadening
As Rob Jeffries comments, gas pressure may also broaden the line profile. The result of this is also Lorentzian, and will hence increase the absorption probability especially in the wings.
A: If we're staying with the purely classical Maxwell equations, they have a certain time reversal symmetry. 
This implies that any system that can radiate on a given frequency, can also receive (in principle) on that frequency.
I cant think of any exceptions to this. 
The same should hold in the quantum realm if we respect further symmetries (namely CPT)
