I would use NSMutableArray, not because it is faster (it probably won't be) but because it is easier in the context of Objective-C. If you use std::stack you will have to add stuff in to do memory management.
I'd then profile the code to find out if the stack was a bottleneck. If it was, I might consider reimplementing with std::stack or even rolling my own.
It is generally accepted that Objective-C is slower in most cases than C++.
Objective-C greatest virtues are flexibility and reuse. The runtime linking that creates that flexibility imposes a considerable overhead.
On the other hand, in the case of arrays, that overhead is usually trivial. In the case of a LIFO stack, you won't see any performance difference between Objective-C and C++ because there code doesn't have to scan the entire array. Unless your array operations are very complex and the arrays very large e.g. 10K+ objects you probably won't see any significant performance differences.
My advice is to do a test run with some dummy data of the type you want to manipulate in the app. Load the arrays up past the max size you expect, loop a large number of manipulations then measure time and memory use. See if the performance gain of C++ justifies the extra dev time and complexity tax is worth the performance gain.
Remember as well that premature optimization is the root of all evil. Don't spend time futzing to prevent a problem you might not even have. Default to the simplest solution unless you have good evidence to suspect it may not work.