If you want the GFI to be part of the breaker in the panel (assumed because you mentioned breaking the tab on the receptacle), it will work if you use a 2-pole GFI breaker (20 amps, using NEMA 5-20R outlets). NEC also requires a common device (e.g. untabbed duplex receptacle) be on a single breaker handle anyway, so you're stuck with the disadvantage of an overload or fault on one causing both to be shut off. A two-pole GFI breaker senses both hot wires and the neutral concurrently (e.g. it has just one current transformer inside).
But there is another option. You can use a pair of GFI receptacles. That will also still work since the GFI current sensor in each receptacle will be comparing the hot and neutral from just its own receptacle connections. The old dryer branch circuit is electrically equivalent to the split-phase service drop coming into your panel. So this means you need a 4x4 two-gang box. But that's what the dryer outlet usually already has. If you have an older one-gang box, you need to expand it to two gang to do this. You can use ordinary duplex GFI receptacles for both phases, but simplex GFI receptacles do exist if you really want just one on each.
If this is an older dryer circuit with no separate ground wire (and had the "sad face" NEMA 10-30R outlet instead of the newer 4-slot one), you will need to disable use of the ground hole. The GFI sensor and test button will still work, since the test button uses the feed-side neutral instead of actual ground.