Just riffing off the concept here...
If photons had charge
-They couldn't be their own antiparticle (as they are now)
-They wouldn't be electromagnetic waves anymore -- since there's no place for charge to be in the wave
-We'd have incredible magnetic forces around *always* (since we'd have lightspeed currents flowing around us)
-Light couldn't always move at light speed because it would have to accelerate in any electric field (right?)
Oh, and just like with the problem of quantum gravity... these bosons would feel their own force. (So just as the graviton creates gravitons because it has mass/energy... and *those* gravitons create gravitons... etc until the universe destroys itself out of sheer exhaustion after making all those gravitons.) A charged photon would give off charged photons would give off charged photons...... so I imagine the forces would diverge / be infinite / please reset the universe and try again.
As for the *range*, having charge shouldn't affect that directly. The reason W+- bosons have a short range is not because they are charged, but because they have mass. (The easy example for this is the W's sibling, the Z0... which is neutral but still short-ranged because of a large mass).
Then again, having a universe completely filled with infinite charged photons would certainly shield the force of any given charge. Not that you'd notice... since there would be literally infinite forces everywhere.
All in all - I'm glad photons don't have a charge.
...and also, I kinda wanna see how Randall Munroe would answer this one as a What If