How much would a SAT oracle help speeding up polynomial time algorithms? Access to a $SAT$ oracle would provide a major, super-polynomial speed-up for everything in ${\bf NP}-{\bf P}$ (assuming the set is not empty). It is less clear, however, how much would $\bf P$ benefit from this oracle access. Of course, the speed-up in $\bf P$ cannot be super-polynomial, but it can still be polynomial. For example, could we find a shortest path faster with a $SAT$ oracle, than without it? How about some more sophisticated tasks, such as submodular function minimization or linear programming? Would they (or other natural problems in $\bf P$) benefit from a $SAT$ oracle?

More generally, if we can pick any problem in ${\bf NP}-{\bf P}$, and use an oracle for it, then which of the problems in $\bf P$ could see a speed-up?
Actually, acceptance of nondeterministic Turing machines in time $t$ is $O(t \log t)$-time reducible to SAT (the construction is via oblivious simulation, see Arora-Barak), so typically any time a nondeterministic machine is appreciably faster than a deterministic one, we'll see at least some speedup with a SAT oracle.

To be more concrete, primality testing comes to mind, as the best variant of the AKS algorithm appears to test primality of an $n$-bit number in time $O(n^6 \; \text{polylog}\; n)$. But if we go "old school", Pratt gave a nondeterministic TM to decide primality in time $O(n^3 \; \text{polylog}\; n)$. Acceptance of this machine can be reduced (deterministically) in $O(n^3 \; \text{polylog}\; n)$ time to a SAT instance.

The 3SUM problem may be another example, as it seems like one can guess a solution and check it in subquadratic time, and then acceptance of such a machine can be reduced to SAT in subquadratic time.