Is it necessary to assume the consistency of $\sf ZFC$ when we prove the completeness theorem? When we use the completeness theorem to show that Peano Arithmetic is consistent, the assumption of consistency of $\sf ZF$ is necessary?
I mean, whenever one says that the completeness theorem prove the consistency of a first-order theory, does he assume that $\sf ZF$ or $\sf ZFC$ is consistent?
Using the completeness theorem, in general, means that we want to equate syntax and semantics. For this we need a framework that can interpret semantics, usually this means set theory such as $\sf ZF$ or its relatives (but not only these!)
Now, you don't have to assume $\sf ZF$ is consistent for using the completeness theorem, because we are working inside $\sf ZF$, that is our meta-theory, as far as Peano is concerned.
But now you might want to ask, should we even use $\sf ZF$ as a meta-theory? Does it prove a contradiction or not? And in that case you need to assume its consistency in the meta-meta-theory, or work with a meta-meta-theory which is strong enough to prove it. But then you need to ask yourself, why is this meta-meta-theory is good enough, and well... it's turtles all the way down from there.
TL;DR No, you don't need to assume anything. But once you start digging into your meta-theory's meta-theory, i.e. the meta-meta-theory, you might want to add these assumptions.