If $S$ is any surface with $h^{1,0}(S) = 1$, then the Albanese morphism is a surjection $a \colon S \to E$ to an elliptic curve, and the pullback $H^0(E,\Omega_E^1) \to H^0(S,\Omega_S^1)$ is an isomorphism (see for instance [this post](https://mathoverflow.net/q/368825/82179)). So the only question is whether the pullback $a^* \omega$ of a nontrivial $1$-form $\omega$ on $E$ has a zero somewhere.

If $a$ is not smooth, then the pullback $a^* \omega$ vanishes at any singular point $s \in S$ for $a$. Indeed, note that $a$ is singular at $s$ if and only if $\dim_{\kappa(s)}\bigl(\Omega^1_{S/E} \otimes \kappa(s)\bigr) > 1$, which means that the pullback $a^* \colon \Omega^1_E\otimes \kappa(a(s)) \to \Omega^1_S \otimes \kappa(s)$ is the zero map (since $\dim_{\kappa(s)} \Omega_S^1 \otimes \kappa(s) = 2$).

So any surface whose Albanese morphism is a surjection $a \colon S \to E$ with at least one singular fibre gives a counterexample. For instance, if $S$ is a surface of general type with $h^{1,0}(S) = 1$ (in this case, the vanishing of its global $1$-form also follows from [PS14]).

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**References.**

[PS14] <cite authors="Popa, Mihnea; Schnell, Christian"> M. Popa, C. Schnell, [*Kodaira dimension and zeros of holomorphic one-forms*](https://doi.org/10.4007/annals.2014.179.3.6). Ann. Math. (2) **179**.3 p. 1109-1120 (2014). [ZBL1297.14011](https://zbmath.org/?q=an:1297.14011).</cite>

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P.S. If you want a more precise grip on numerical invariants of surfaces, there is a great tool by Pieter Belmans and Johan Commelin with loads and loads of data. See *Le superficie algebriche* at [https://superficie.info](https://superficie.info).