Edit: I found a better example. Consider these clauses:
& \lnot P(x) \lor P(f(x)) \\
& P(x) \\
& \lnot P(f(f(x)))\\
Obviously, this set of clauses is contradictory. But without renaming variables, the only possible resolvent is $P(f(x))$ and no more resolvents are possible - all lead to substituting $f(x)$ for $x$, which is impossible.
Edit: Consider the meaning of clauses. Each clause is implicitly universally quantified. So the meaning of its variables isn't fixed to anything. Now let's say you have two clauses both containing $x$. If you perform resolution without renaming $x$ in one of them, then you add a meaning to $x$ which it doesn't have: you say that $x$ means the same thing in both clauses, which is not true.
If you don't have distinct variables in your clauses, resolution will give you too weak conclusions.
(The original answer.)
For example, let's have 4 clauses:
- $A\lor B(x)$
- $\lnot A\lor C(x)$
- $\lnot B(c)$
- $\lnot C(d)$
where $x, y$ are variables and $c, d$ constants. If we perform resolution on the first two without renaming $x$, we'll get $B(x)\lor C(x)$. We can proceed with $\lnot B(c)$ to get $C(c)$ but now we cannot resolve it with $\lnot C(d)$.
On the other hand, if we rename $x$ to $y$ in the second one to have disjoint set of variables, we'll get $B(x)\lor C(y)$ from the first resolution step and we can derive an empty clause using $\lnot B(c)$ and $\lnot B(d)$.