As has been pointed out in other answers, the basic diode snubber arrangement extends the relay drop-out time and the speed of contact opening.

simulate this circuit – Schematic created using CircuitLab
Figure 1. (a) Diode snubber. (b) Resistor snubber. The diode can be left out but doubles the power consumption of the circuit.
I checked a 24 V Finder relay in my stock and measured 800 Ω coil resistance and 2.8 H inductance de-energised and 9 H when the armature was pressed in.
At 24 V the energy stored in the relay would be \$ \frac {1}{2} LI^2 = \frac {1}{2} L{\frac {V}{R}}^2 = \frac {1}{2} \frac {24}{800}^2 = 4.5 \ \text {mJ} \$.
We know from the maximum power transfer theorem that maximum power transfer occurs when the load resistance (R1) is equal to the source resistance (the 800 Ω of the relay coil). The reader can prove to their own satisfaction by calcuus that the maximum time delay will occur when the relay is short-circuited at the instant of power-cut. (For simplicity assume that D1 is ideal.)
Running this through a simulation results in the graphs below.

Figure 2. The switches are opened at t = 1 ms. The time to drop to half-current for circuit 1a is shown by the blue line and occurs 7.3 ms after the switch opens. The time to drop to half-current for circuit 1b is shown in orange and is 3.8 ms.
The relay will drop out at twice the speed for the resistor snubber.
... MOV instead of a flyback diode with a relay coil. ... but why would you want to do that?
If the MOV can be rated to break down just above the relay voltage then it may be close to the optimum relay load to minimise drop-out time. It may behave faster than a diode and uses one component instead of two as shown in Figure 1b.
The only reason I can think of is that the coil is normally energized and seldom de-energized, and the MOV acts as a speed-up circuit for de-energization, burning the inductive energy faster than a flyback diode.
I agree with that.
That said, why not use a zener or a resistor, especially given that MOVs have limited lifetimes? What other options are there?
I think we've covered this.