Faster switchoff? What on earth are you talking about man ... There is no way in hell to get a mechanical relay to switch off any faster, other than to remove the power to it.
Of course you need to remove the power. But the the collapsing field of the inductor will keep the current going. The energy stored in the inductor needs to be dissipated somewhere.
Initially, right after the power is cut off, the current will be the same as before, but going round in the loop with the flyback diode. The diode dissipates energy according to forward voltage*current. A resistor in place of the diode can drop potentially drop more voltage, thus dissipate more energy, thus collapse the magnetic field of the inductor faster, potentially leading to a faster release of the relay.
The obvious disadvantage is that if not combined with a diode, the resistor will dissipate energy during the whole switch on time of the relay, so the resistor needs to be able to take that energy and the supply needs to be able to supply the additional current. It is certainly not advised in everyday situations. Also, the bigger the resistor is, the higher the voltage created by the inductor will be, so you need to be careful with the breakdown voltage of the mosfet.
Here is an interesting stackoverflow article about different methods to deal with the collapsing field energy.
Now, don't ask me when it would be sane not to add a diode to the resistor, but I think I have heard of such applications, they are certainly possible and in principle do the job. It is certainly a very special case. Or maybe I am not remembering correcly, and there was always an additional diode that was just left out in some discussion about the field breakdown because it made a minor contribution.