We show that if a Laurent series $f\in\mathbb{C}((t))$ satisfies a particular kind of linear iterative equation, then $f$ is either an algebraic function or it is differentially transcendental over $\mathbb{C}(t)$. This condition is more precisely stated as follows: We consider $R,a,b\in \mathbb{C}(t)$ with $R(0)=0$, such that $f(R(t))=a(t)f(t)+b(t)$. If either $R'(0)=0$ or $R'(0)$ is a root of unity, then either $f$ satisfies a polynomial equation, or $f$ does not satisfy a polynomial differential equation. We illustrate how to apply this result to deduce the differential transcendence of combinatorial generating functions by considering three examples: the ordinary generating function for a family of complete trees; the Green function for excursions on the Sierpinski graph; and a series related to the enumeration of permutations avoiding the consecutive pattern 1423. The proof strategy is inspired by the Galois theory of functional equations, and relies on the property of the dynamics of $R$, Liouville-Rosenlicht's theorem and Ax' theorem.