I believe, and this is foggy, that the answer is that the majority of the compliance in a driver is the spider on the inside, next to the magnet. The foam or rubber ring we see on the outside is normally the one that contributes the least amount.
I would agree with this. The weight of the entire driver plus the weight and the "spring" strength of the spider will dictate the resonance frequency of the woofer. This applies to port size tuning. I don't think rubber/foam surround will affect the weight or spring/flex of the driver and have very little impact on resonance frequency.
Now the type of surround can definitely impact much higher frequencies (such as anything above 400-500 hz) because it affects the natural damping of the speaker vibration resonance. This may have impact on frequency response or crossover design. If it's a woofer on a 3-way speaker, then I wouldn't worry about replacing surround with rubber.