Typically the entanglement between photons is with respect to their polarizations. As long as the interaction between a photon and a mirror is polarization-independent, it should not affect the entanglement between the photon and the twin photon with which it is entangled. However, it is safe to conclude (and pretty easy to test) that if the mirror is polarization-dependent it will reduce the (polarization) entanglement between the photons. If entanglement is with respect to another property of the photons such as the direction of propagation, phase, or frequency, then the mass of the mirror will enter into the equation: quantum uncertainty in the velocity of even a tiny mirror whose mass is a few grams is negligible, so the amount by which reflection can add uncertainty to the frequency of a reflected photon (and thereby reduce frequency entanglement between it and its twin) is negligible.
Not that Wikipedia should always be trusted, but in this case it is correct [https://en.wikipedia.org/wiki/Quantum_entanglement]:"Measurements of physical properties such as position, momentum, spin, and polarization, performed on entangled particles are found to be correlated." The point there is that any conserved quantity can lead to entanglement: when two particles are created, e.g., by conversion of a gamma ray into an electron-positron pair or conversion of a visible light photon into two new photons of twice the wavelength of the original via downconversion, the two new particles are entangled. Quantities that are conserved (energy, momentum, spin, charge, etc.) result in correlations between the corresponding properties of the new particles, such that the total values of those conserved quantities are not changed by the creation of the new particles. Measuring the value for one particle automatically reveals the value for the other particle.
Loss of entanglement or decoherence results when any part of the value of a conserved quantity (let's use momentum for convenience) is transferred to the outside systems. The total momentum is still conserved, but only a portion of it is represented by the sum of the two particles' momenta. The rest is somewhere off in the rest of the universe; and to the extent that the amount of that quantity that is lost cannot be determined, the correlation between the momenta of the two particles is reduced.
In the case of one of a pair of entangled photons reflecting off a mirror, the interaction between the mirror and the photon cannot alter the energy of the photon significantly. It does alter the momentum of the photon of course, but in a very well-determined way simply because the mirror is, from the photon's perspective, immovable. So whatever correlation there was between that photon and its twin is not lost; it is simply altered in a predictable way. The same is true for a photon's polarization: circular polarization reverses sign upon reflection from a mirror, so the mirror alters the polarization of a photon, but in a fully predictable way.
On the other hand, if the mirror has extremely low mass, comparable to an atom, it will recoil upon reflecting the photon. The reflected photon will have an unpredictably altered frequency and momentum simply because the exact position, orientation, and velocity of the mirror are subject to quantum indeterminacy. As a result, there will be a loss of entanglement coherence: measurement of the photon's properties will not provide a fully reliable basis for predicting the properties of the twin photon.
However, in setups of the sort you are concerned with, the mirrors are massive and, from the photon's perspective, immovable. Yes, there is loss of coherence upon reflection, but the percentage loss is immeasurably small.