What is the danger of a bootleg ground on a 110 outlet wired from a 3 wire 240 circuit?
Neutral is not ground.  Neutral is the desired current return, ground is a safety device only. 

Let's look at some fault conditions, and we'll lead up to the problem with the bootleg ground.  First, consider a broken neutral wire (near the service panel) with no loads on the circuit whatsoever.   



Nothing is connected to the neutral; it's floating.  That's not the same as being at zero.  Nothing pulls it down to zero (neutral) but nothing pulls it up to hot either.  

But what if something did?



This is concerning.  You see the broken neutral is being pulled up to 120V.  Not toward 120V - all the way, because absolutely nothing pulls it down to real neutral.  This is deadly if you have a NEMA 10 dryer or stove circuit, where they intentionally attach chassis to neutral as a substitute for ground.  But other than that, this situation is not that hazardous if everything is perfect and to code, and probably wouldn't even trip a breaker.  And since neutrals stay with their mated hots, this problem is contained to a single circuit.  

Now to your hypothetical: a bootleg ground.  Same, but substitute ground for neutral.  



Holy smokes!  We've now lit up every ground downstream of the break.  Unlike neutrals, grounds may be cross-connected, so it can affect every circuit.  Every screw in every outlet cover, any metal conduit, any appliance which properly grounds its chassis (particularly NEMA 14 connected dryers and stoves), the exterior of your metal power strips, maybe even the chassis of your PC. 

In fact a single point of failure - the ground-neutral bond in your main panel - combined with a bootleg ground, will create the worst case scenario: energizing every ground in your house including the service panel itself. 

Other than that, I see ground breaks all the time - especially in EMT conduit, where those flimsy zinc couplers can snap off. 

Here's why the grounding rod won't help.  Dirt is a lousy conductor, except when it's not.  Grounding is done to force the house's ground system (and neutral) to be near earth, to reduce shock hazard.  It's not meant as a current return, and it works so badly as one, that using it that way creates the hazard!



Power wants to get back to neutral, not ground.  The only place ground can reach neutral is back at the transformer pole, and that current path goes through unreliable dirt. Code says a single ground rod must be 25 ohms (but if you have two, you don't even have to test!  This is because of the practical difficulty of getting good earth connections in some places.)  Run the math:  E=IR, 120V=4.8A*25Ω.  In other words a dead short would flow 4.8 amps.  That's not going to trip a breaker, and can't be counted on to reduce shock hazard from that bootleg ground.  The bootleg load could be considerably less than 25 ohms, so it's going to pull ground up toward 120V better than earth may pull it down.

I know it's a busy drawing, click to zoom. Here's what else is going on there: High voltage 3-phase delta (yellow orange brown) feeds a single-phase neighborhood transformer, whose secondary is 240 volts center-tap. The center tap is Neutral, and it's bonded to a grounding rod at the transformer. Those 3 wires, hot-neutral-hot, are bundled together and carried to each house via its meter.  Each house has a grounding rod bonded to ground, which is also bonded to neutral in the main panel.