Why can't gauge bosons have mass? Clearly, a mass term for a vector field would render the Lagrangian not gauge-invariant, but what are the consequences of this? Gauge invariance is supposed to be crucial for the renormalisation of a vector field theory, though I have to say I'm not entirely sure why.
As far as removing unphysical degrees of freedom - why isn't the time-like mode $A_0$ a problem for massive vector bosons (and how does gauge invariance of the Lagrangian ensure that this mode is unphysical for gauge bosons)?
 A: Let me anwser a closely related quenstion:
Consider a U(1) gauge theory with massless gauge bosons, can any small perturbations give the gauge boson an mass. 
Amazingly, the anser is NO. The masslessness of the gauge boson is topologically robust.
No small perturbations can give the gauge boson an mass.
For detail, see my article.
Let me make the statement more precise. Here we consider
a compact U(1) gauge theory with a finite UV cutoff (such as a lattice gauge theory),
that contains gapless gauge bosons at low energies.
Then no small perturbations to this compact U(1) gauge theory with a finite UV cutoff 
can give the gauge boson an mass, even for the small perturbations that break the gauge invariance.
So the masslessness of gauge boson is a stable universal property of a quantum phase.
Only a phase transition can make the gauge boson massive.
A: Sure gauge bosons can have mass. The Higgs mechanism for one. Or in 2+1D, add a Chern-Simons term. Or in 3+1D, and a two-form gauge field B with the gauge symmetry $B \rightarrow B +d\lambda$ with $\lambda$ being any 1-form, and add a $B \wedge F$ coupling term.
If you mean why they can't have mass in the Coulomb gauge, the answer is topological. Look at the dispersion relation with helicities. A mass gap translates into an energy gap which would split the light cone dispersion into hyperboloids. Not possible topologically.
A: As far as renormalizability of gauge bosons, a nonabelian gauge theory with massive gauge bosons is non-renormalizable as I describe here: What evidence is there for the electroweak higgs mechanism? . A massive abelian gauge boson is renormalizable and to my knowledge suffers no such defects at high energy as the would-be goldstone boson is from a linear sigma model, not a non-linear sigma model.
As far as the unphysical time-like modes, while there is no gauge invariance, what really matters is that its coupled to a conserved current, so there is still the cancelation that happens with massless gauge bosons.  
