Consider the unital $C^*$-algebra $C([0,1], \mathbb{C})$, and its positive elements $f : x ↦ \sqrt{1-x^2}$ and $g : x ↦ x^2$. Then we have, for all $x ∈ [0,1]$:
$$(fg)^*(x) = g^*f^*(x) = g^*\left(\overline{\sqrt{1-x^2}}\right) = g^*\left(\sqrt{1-x^2}\right) = \overline{1-x^2} = 1-x^2 ≠ \sqrt{1-x^4} = f(x^2) = fg(x).$$
Thus, as $fg$ is not self-adjoint, it is not positive. Have I found an example of a unital $C^*$-algebra $A$ with positive elements $f, g$ such that $fg$ is not positive? I would say yes, were it not for Example VIII.3.2 in Conway's book, which says an element $h$ of $C(X)$ is positive if and only if $(\forall x ∈ X)(h(x) ≥ 0)$, which is certainly the case for $h = fg$... (In fact I used this property to say that $f, g$ were positive in the first place..)