Q:

Mass corrections to fermions proportional to the mass? 

In this post regarding quantum corrections to a massless fermion field, the answerer stated that quantum corrections to the mass will always be proportional to the mass (at least in QED).  This point leads to an important claim that a massless fermion remains massless under quantum corrections in the Standard Model. 
I can show that this is true to 1 loop order in QED, but is this always the case?  If so then why? 

A:

It can be seen to follow from a more general statement, namely: "if a parameter in the theory is such that the symmetry gets enhanced when it vanishes, then at every order in perturbation theory the corrections to this parameter will be proportional to its bare value".  
This is because perturbation theory respects the symmetry of the classical theory.  If the bare parameter is zero, there is an enhanced symmetry that is respected order by order in perturbation theory, and therefore the symmetry breaking parameter never gets generated.  It follows that every correction in the theory with bare parameter $\neq 0$ must be a function of the parameter that vanishes when evaluated in $0$.
In the case of the fermion mass, the enhanced symmetry is the chiral $U(1)$.
This general statement is the reason why a parameter can be "naturally small" whenever setting it to zero results in enhanced symmetry.