Do any major theorems of complex analysis that require holomorphic functions fail if the function is only holomorphic up to removable singularities? Or is "holomorphic on $\Omega$" universally (i.e., in practice and standard texts, and including for purposes of Ph.D. quals) understood to mean "at worst possessing a holomorphic extension to $\Omega$"?  To be sure, I can think of extremely trivial, sub-pedantic examples of statements that would require a strictly holomorphic function like "let $f$ be entire, then $f$ is defined everywhere."  Yes, this "theorem" fails for $f(z) = z/z$ at the origin because of the hole, but it hardly gives useful information (if it forms the cornerstone of your current research, however, please accept my sincere apologies).
What I'm more concerned about is assuming that it's OK to have the "weaker" definition somewhere that the "stronger" definition is actually called for, e.g., when applying one of the major grad-level-course theorems.  I can't think of any specific instances in which the distinction is relevant, but it crossed my mind as I tried to prove that the convergence of the Taylor series of $f$, holomorphic at $z=c$, on a disc $D(c, r)$ implies the holomorphicity of $f$ on that disc.  That implication is supposed to be true, but it is not true if removable singularities cause a function to not be holomorphic; indeed, all we have to do is replace $f(z)$ with $f(z)\frac{z-w}{z-w}$ for some $w$ in $D(c, r)\backslash\{c\}$ for a counterexample.  
I just don't want to lose generality in an argument by taking $f$ with removable singularities up to its analytic continuation.

A bit less trivial (but still trivial) example is Picard theorem
