Well you're not going to prove 0=1 in PA, because PA is consistent,
(though not PA-provably so), following Gentzen. But I digress.

If you proved 0=1 in, say, ZFC, that would simply mean that
ZFC was inconsistent - that the entities it purported to
describe had no reasonable interpretation and that logical
conclusions derived from the axiom had, in general, no bearing
on the world.  In particular, it would be irrelevant that you
had proved P = NP.  But I still digress.

My main point:  your 0=1 proof is a purely combinatorial object -
a symbol sequence that satisfies syntactic constraints that
can be checked in polynomial time.  The standard Zero-Knowledge
Proof technology would apply to this proof just as to any other.
The cataclysmic semantics of the proof's conclusion would simply
be irrelevant.  

Surely if ZFC turns out inconsistent, much of set theory could still 
be saved by suitably weakening say, the particular axiom whose 
self-evidence turned out illusory.  (Consensus in the short 
term concerning which axiom to give up might turn out difficult 
to achieve).  At the end of the day, the offending axiom would
simply seem overambitious, just as the occasional large cardinal
axiom turns out to be a turkey, roadkill on the transfinite superhighway
if you will.  Most of classical mathematics will still go through intact,
and the theory of finite sets, PA essentially, already strong enough
to articulate the P=NP conjecture, will remain consistent.