If there's no mass then E=cp (from E^2 = c^2 p^2 + m_0^2 c^4)
So you've still got a problem with infinities
You've asserted that Energy is still conserved so E=hf should still hold (for a photon). Assuming Planck's constant doesn't change then \lambda must become infinite if c becomes infinite which, in turn implies that the universe must be infinitely large.
The problem with all these hairbrained schemes is that people throw them around without working through all the consequences and explaining exactly how they are all dealt with.
When that is done it's almost always the case that there's something apparent that we already know to be false.
(I'll leave it as an exercise to see what happens if Planck's constant also changes