The core problem is: Maps of symmetric spectra are, levelwise, maps equivariant with respect to the symmetric group, but the notion of weak equivalence ignores that.
The injectivity notion essentially works because there is a (simplicial) model structure on symmetric spectra where
weak equivalences are levelwise weak equivalences,
cofibrations are levelwise monomorphisms, and
fibrant objects are "injective" spectra.
This is sometimes called the injective level model structure. This ensures that $Map_{Sp^\Sigma}(X,E)$ has a sensible homotopy type, making $\pi_0$ invariant under levelwise weak equivalence. The $\Omega$-spectrum condition is not as relevant.
However, you've asked for an example, and so we have to come up with one. The problems with doing so are as follows.
There is more than one way to skin this particular cat. There are at least three different level model structures on symmetric spectra (see Stefan Schwede's book project, chapter III, for coverage of them), and if we try to come up with an example we need to have one where the mapping space has an incorrect homotopy type for all of them.
We need $E$ to be an $\Omega$-spectrum. Since this implicitly involves loop spaces, I want to stick to an example where $E$ is levelwise fibrant (and hence is fibrant in the projective model structure). So we need an example where the domains are not projective-cofibrant, which boils down to a condition about the symmetric groups not acting sufficiently freely.
I am lazy, and want to construct the domain spaces using a "free" functor so that I can analyze the mapping spaces easily. There are the functors $G_n$ described in Hovey-Shipley-Smith's paper, which are left adjoint to the forgetful functor $X \mapsto X_n$ from symmetric spectra to $\Sigma_n$-spaces. Objects constructed in this way are always cofibrant in the flat level model structure, so we need the target $E$ to not be fibrant in the flat level model structure. This is a specific condition on what the fixed-point subspaces of $E_n$ look like.
Again in the "laziness" department, you've asked for an example where $E$ is an $\Omega$-spectrum, and these are more difficult to write down.
Thus, here is a recipe for taking some injective $\Omega$-spectrum $E$ and constructing a new one, $E'$, in a way ensuring that $E^0$ doesn't take weak equivalences to isomorphisms.
Let $X \to Y$ be the map $$G_n((E\Sigma_n)_+ \wedge S^k) \to G_n(S^k),$$ induced by the projection, where $S^k$ is given the trivial action of the symmetric group. One checks by the explicit formulas for $G_n$ that the map $X \to Y$ is a levelwise equivalence. For any symmetric spectrum $E$, the map $E^0(Y) \to E^0(X)$ is $$\pi_0 Map_{\Sigma_n,*}(S^k, E_n) \to \pi_0 Map_{\Sigma_n,*}((E\Sigma_n)_+ \wedge S^k, E_n)$$ which is $$\pi_k ({E_n}^{\Sigma_n}) \to \pi_k Map_{\Sigma_n}(E\Sigma_n, E_n).$$ If $E$ is injective, this is an isomorphism by assumption.
Let $$E_n' = E_n \wedge (E\Sigma_n)_+.$$ Here the symmetric groups act diagonally, and the structure maps are induced by the structure maps of $E$ and the functorial maps $E\Sigma_n \to E\Sigma_{1+n}$. Then $E'$, levelwise, has no fixed points, and the map $(E')^0 (Y) \to (E')^0 (X)$ is $$ 0 \to \pi_k Map_{\Sigma_n} (E\Sigma_n, (E\Sigma_n)_+ \wedge E_n) \cong \pi_k Map_{\Sigma_n} (E\Sigma_n, E_n) $$ by freeness. If $E$ was injective in the first place, the right-hand side is equivalent to $\pi_k E_n$. If $E^*$ is not trivial, we can pick some $k$ and $n$ making this nonzero.
