How can philosophy deal with the nonphysical in light of scientific
Primarily, respect for obtaining knowledge (read: philosophy) contends with the non-physical by recognizing the distinction between the physical and the non-physical (see "the excluded middle"). Epistemic limits do not preclude speculation and supposition, nor do they render axiom and self-knowledge invalid, nor language meaningless.
Logically, if the only proof of existence we have is what we can see,
hear, smell, touch, taste, and feel the movement of, [then] why do many
philosophies deal with issues outside the physical?
Proof is merely a means of evaluating conjecture as truth. If by "philosophies" you mean "ways of looking at things" then I can only offer that if philosophy were a way of looking at things, then an oasis and a mirage would have epistemic and ontological equivalence. They do not.
Due to these scientific limitations of not being able to prove the
spiritual exists, why doesn't philosophy eliminate these concepts?
What is meant by 'spiritual'? If, as etymology suggests, you mean "breath" and both the conscious as well as autonomous functions by which we breathe without suffocating while we sleep, then the purview for inquiries is biology. If you mean self, then there is psychology, neurobiology and the philosophy of mind to address your concerns regarding spirit. If you mean a transcendental or immortal soul, you have religion and whatnot. It is not the task of philosophy to eliminate concepts, instead the virtue of wisdom can help focus your investigations by understanding what you are investigating. Hope that helps!