Classes with oracles forcing the caller to immediately accept (or reject)? Usually an oracle operates as a black box, simply returning an yes or no answer to the machine calling it; once it has the result, the caller is free to proceed as it sees fit.

However, what happens if the caller must accept or reject (exactly one of the two) immediately as a function of the oracle's output? Say, for example, the caller has access to a SAT oracle, but must always accept (unconditionally) whenever the oracle answers with "yes"; this means the caller can continue its operation if and only if the query formula is unsatisfiable.

This imposes a restriction, obviously; are there cases in which this yields — or there is reason or evidence enough to suspect it yields — a strict subset of decision problems when compared to the "usual" behavior?

I am particularly interested in the case of $\mathsf{P}^{\mathsf{NP}}$ or similar classes in the polynomial hierarchy, but other classes and oracles are relevant too.
The question is not entirely clear to me. However, concerning the example that is spelled out more precisely: if a language is recognizable by a poly-time machine with a SAT oracle which must accept whenever the oracle answers “yes”, it is in fact in NP. First, observe that regardless of the oracle answers, we can simulate in polynomial time the only possible run of the algorithm by pretending all answers are “no”. (In particular, we can compute the list of all oracle queries beforehand.) Then, the original algorithm accepts iff one of the oracle answers in the simulated run is actually “yes”, or the simulated algorithm accepts in the end. This is an NP property.

More generally, if we use an arbitrary oracle $A$ in the above scenario in place of SAT, a language can be computed in the indicated way iff it is poly-time dtt-reducible to $A$.