If a pulse is reflected by a mirror that absorbs part of its spectrum, must it become longer? There is a pulse of duration $T$ supported by a certain bandwidth $\omega$.
The pulse is reflected by a mirror that absorbs part of the spectrum.
Does the pulse become longer?
Is there a situation when the time-bandwidth product does not hold?
 A: 
There is a pulse of duration $T$ supported by a certain bandwidth $\omega$.
The pulse is reflected by a mirror that absorbs part of the spectrum.
Does the pulse become longer?

Generically, yes, but not necessarily.
As a simple counter-example would be a superposition of a Fourier-limited pulse centered at $\omega_0$ with bandwidth $\Delta \omega$, together with a second pulse with disjoint spectrum of equal bandwidth and centered at $\omega_0+2\Delta\omega$, such that the second pulse is strongly chirped. If this joint pulse is reflected through a mirror that is reflective around $\omega_0$ and completely absorptive around $\omega_0+2\Delta\omega$, then the strongly chirped (and therefore temporally longer) component is gone, and you're left with a pulse that's shorter than the original.
Of course, this relies on having an original pulse which was far from saturating the time-bandwidth inequality.
And, of course,

Is there a situation when the time-bandwidth product does not hold?

No, there isn't.
