How precise must the energies match for absorption of photons?

In atoms the energy levels do not have a precise energy. When you solve Schrodinger's equation for an atom the results are the energy eigenfunctions. However these are functions that are time independent, and they have an exact energy only because they are time independent.

At the risk of oversimplifying, you can regard this as an example of the energy time form of the Heisenberg uncertainty principle:

$$ \Delta E \Delta t \ge \frac{\hbar}{2} $$

If $\Delta t$ is the lifetime of a state then $\Delta E$ is the uncertainty in the energy of that state. For the energy eigenfunctions $\Delta t = \infty$ so $\Delta E = 0$ and the energy is precisely defined.

The point of all this is that in an atom an excited state has a finite lifetime and therefore it has a finite energy uncertainty, and this produces an effect called lifetime broadening. This means transitions to and from the state can occur for photons with a range of energies. The range of energies allowed depends on the energy uncertainty of the state, which in turn depends on its lifetime.

Agree with the above, but also if the atom, or collection of atoms, are in thermal equilibrium, then there is another broadening mechanism, besides lifetime broadening, called Doppler broadening that accounts for the motion of the atom(s). This has the effect to substantially widen the effective line width depending upon the temperature.