Well, we must be careful about what 'mathematics' means here. We generally mean some kind of formal system (the things Godel's theorem talks about) capable of formalizing and proving theorems of interest. Certain kinds of mathematics require stronger systems than others, and thus are at a higher risk of being inconsistent.
Godel's theorem itself is about formal systems, which consist of strings of symbols that we manipulate syntactically to form and verify proofs. This is very simple reasoning at its core: a computer program can verify if something is a valid formal proof or not. As such, Godel's theorem can be talked about in very simple formal systems. In fact the system PRA (which is commonly associated with purely finitary reasoning) is sufficient to prove Godel's theorem$^*$.
In order for us to believe in the meaningfulness of Godel's theorem, we need to believe that PRA is a sound and consistent system. Fortunately, it's such a weak system, that almost everyone, even extreme skeptics, believe it is consistent. However, PRA is powerful enough arithmetically for Godel's theorem to apply to it, and thus we cannot prove the consistency of PRA in a weaker system than it. Ultimately, it must be assumed.
Remember, if you believe a system is correct (i.e. it talks about 'real' mathematical objects, whatever that means, and only proves true things about them) then you believe it is consistent. So if we believe in natural numbers - not the completed infinite set of natural numbers, just natural numbers - and that certain statements about natural numbers have meaning (namely, the quantifier free statements about primative recursive functions that PRA can talk about), and that PRA is a valid way about reasoning about them, then we believe PRA is consistent. As I said, this is something most people believe even at their most skeptical. (Though not universally.)
But not all math is a simple matter of string manipulation and finitary reasoning. Much of modern math requires thinking about infinite collections of objects. Traditionally, math has been considered to be founded on set theory, with the ZFC axioms and first order logic as our formal system of reasoning. This is a much stronger system then PRA. For instance, it easily proves that PRA is correct and consistent. There is nothing contradictory about this since it's stronger. And of course it's a strong enough arithmetically that Godel's theorem applies to it, so it cannot prove itself consistent.
So if we're taking 'mathematics' to be ZFC, then 'mathematics' (actually a very weak fragment of 'mathematics') proves Godel's incompleteness theorem, and thus 'mathematics' proves that it cannot prove itself to be consistent. Thus we will have to assume the consistency of ZFC if we want to take its proofs seriously (or, more exactly, whatever fragment of ZFC we use in the proofs).
So, basically, that's a big yes. We need to make assumptions about consistency, in order to do any math at all, and the more math we want to do the stronger assumptions we're going to need to make. (And see Asaf's excellent answer about just how far this goes and how precise this is.)
A few more comments.
- First, there's kind of a weird fixation on theories proving
themselves consistent. Recall that it's a basic tenet of classical
logic that if a system can prove an inconsistency, it can prove
anything. So if you doubted the consistency of a system, why would
you believe its proof that it's consistent? If it were inconsistent,
it could prove that too. Really, the dream scenario would be something like PRA that's very weak with
little philosophical commitment proving something very strong like
ZFC to be consistent. Of course Godel dashes our hope here too: anything provable in the weaker system would be provable in the stronger system too. However, it wasn't an unreasonable request at the outset. Consistency is just a statement about syntactic manipulation of symbols, so it doesn't require a very sophisticated system to talk about. The consistency of ZFC is expressible in PRA, even if it's not provable there.
- Second, actually, ZFC can prove every finite fragment of ZFC is
consistent. Thus with any ZFC proof (which must use only a finite
number of axioms) we have a proof in ZFC that the axioms we used
were consistent. This is not super comforting for the reasons stated
in my previous comment. And what of the axioms we used in our proof
that the finite fragment was consistent... were they consistent?
- Third, while Godel says we can't prove the consistency of a system
from a weaker one, there's a famous loophole there. Peano arithmetic
is a much-studied system of arithmetic that seems to delineate
'purely arithmetical' (as opposed to analytical, or set theoretical,
etc) reasoning. It can be proven consistent in ZFC, but as I've
emphasized, it's of little interest that set theory proves
consistency if you're wondering if maybe something's fishy about
arithmetic. However, Gentzen came up with a famous proof that
uses mostly very weak finitary reasoning about formulas (just
PRA, in fact, which is weaker than PA). However, it assumes that transfinite induction
through the ordinal $\epsilon_0$ is valid. This is much less scary
and infinitary than it probably sounds, but it's still something
that can't be proven in PA, in accordance with Godel. So PA can be
proven consistent in a system that's a lot weaker than PA in some
ways, while being a little stronger in another.
- It is a logical possibility that ZFC is consistent, but it proves
itself inconsistent. The key to understanding this is that a system
can be consistent, but also be wrong. 'ZFC is inconsistent,' as
represented in ZFC, is an arithmetical statement about the existence
of a number that encodes the proof of a contradiction. So more
precisely, ZFC would be arithmetically unsound... it would prove a
number with a certain property exists that doesn't actually exist.
What's more strange, whether a number is a proof of an inconsistency
is a very simple computable property - so simple that ZFC is bound to get it
right - so for any number you name, it would prove that it doesn't
have that property (even though it proves there exists a number with
that property!). This seeming, but not literal, inconsistency is
called $\omega$-inconsistency. This would mean that ZFC (more
precisely, all models of ZFC) does not actually have the natural
numbers we know and love, but rather has a set of nonstandard
integers. This is a structure that can't be distinguished from
the natural numbers by first order logic, but consists of both the
standard naturals and 'infinite' elements larger than all the
naturals. The proof of ZFC's inconsistency would be one of these infinite
$^*$I'm not one hundred percent positive that PRA suffices for the incompleteness theorem as stated here. It's just what I've heard. In any event, it works in a very weak system, certainly much weaker than Peano Arithmetic.