Why do photons add mass to a black hole? Why do photons add mass to a black hole?  
When photons are taken irreversibly into a black hole does the mass of the BH increase?
 A: This is really just an expansion of Graham's answer.
It's a commonly made mistake that gravity, and therefore a black hole, is caused by matter. In fact the spacetime curvature is related to a quantity called the stress-energy tensor. This is usually represented by a matrix with ten independant values in it (it's a 4x4 matrix but it's symmetric so six of the elements in it are duplicated).
Only one of the elements in the matrix, $T_{00}$, depends directly on the mass, and actually that element gives the energy density, where mass is counted as energy using Einstein's equation $e = mc^2$.
So photons affect spacetime curvature because they contribute to the energy density even though they have no mass. Actually photons contribute to other elements of the matrix as well because they have a non-zero momentum and this too affects the spacetime curvature.
Re your question:

When photons are taken irreversibly into a black hole does the mass of the BH increase?

Yes, the mass of the black hole will increase by the photon energy divided by $c^2$.
Re your comment to Graham's question, yes, provided you add more energy than the black hole is radiating you will maintain or increase the black hole. You could add the energy using lots of low energy photons or a few high energy photons. It's the total energy added that matters.
A: Although photons are rest massless and the Einstein equation E=mc^2 isn't appropriate here they still have massenergy corresponding to E=plancks constant multiplied by the photon frequency.
Black holes do have a temperature, very low for astronomical BHs, so all photon energies above those of the cosmic background radiation temperature at 2.725 Kelvin, incident upon the BH will slowly increase its mass.
