An alternative to considering the law of the excluded middle as an axiom is to consider it as a definition. You can consider the definition of a Boolean value to be that it is either true or false, then the mechanic of the logic becomes to simply to determine whether we can prove that a value is a Boolean, at which point you can deduce the excluded middle.

But you asked for an example. Here is one I came across once.

There is a duality between sets and functions. For example, one person may write $S \cup T$ and another may write $s(x) \lor t(x)$. It is a worthwhile exercise to convert expressions between their set form and their function form. So let's do that with Russell's self contradictory set:

Set form:

$$P \equiv \{x \mid x \not \in x\}$$

Converted to boolean logic and functions becomes:

$$p \equiv (\lambda q)\, \lnot q(q)$$

Russell's paradox comes from considering $P \in P$. The function form equivalent is to consider $p(p)$:

$$p(p) = \bigg((\lambda q)\, \lnot q(q)\bigg)(p) = \lnot p(p)$$

Is $p(p) = \lnot p(p)$ paradoxical? No, because we haven't defined that $(\forall x )p(x)$ must be a boolean. We haven't assumed the excluded middle. On the other hand, some logics do assume $(\forall x,y)\,x\in y$ is a boolean, which does assume the excluded middle (and make a heroic attempt to limit set comprehension), which does result in the definition of $P$ being paradoxical.

There are trade offs in the design of a logic. If you only use first order logic, you can assume the excluded middle all day long. If you want to use higher order logic and partial logics (logics where the domain of functions isn't the universe), then you give up the excluded middle.

Another way of looking at this question (there seem to be many) is in terms of decidability.

Godel established that in every sufficiently description axiom/inference set, either there is a grammatically valid statement that is undecidable or that your logic is self-contradictory.

Now what happens if we assume every undecidable statement $D$ is either true or false? Let $A$ be the set of all possible assignments of true or false to $D$:

$$\forall d \in D ~~\bigg(d \lor \lnot d\bigg)$$
$$\exists c \in A ~~\forall d \in D ~~ \bigg(d = c_d\bigg)$$

Still being fairly informal, the law of the excluded middle implies that there is at least one assignment to the undecidable statements. But Godel established that at least one undecidable statement must exist which is neither true nor false for the axiom set to be consistent. I'm fairly certain that this inevitably leads to a paradox, although given all the encoding associate with the Godel Sentence it might be very complicated and roundabout.

Either way, it is just easier (from a consistency POV) to not assume that all grammatically correct propositions are true or false, even if it does make some proofs harder or nonexistent.