I want to say something less trivial, so I would like to propose an approach to this question in the case when the Riemann surface $M$ is a punctured Riemann surface (without boundary) of finite genus. I claim that such an example doesn't exist. This goes in the direction of what the question is asking for.

*Proof.* The main idea of the proof is that in the situation as I've described, if we take the pull-back $f^*(g)$ of the hyperbolic metric on $\mathbb H$, the metric $f^*(g)$ is a hyperbolic metric with cusps and a finite number of conical points of angles $ 2\pi n$ ($n\in \mathbb Z_+$) on $M$. I will justify this statement later on, but first I want to say how to get a contradiction from it.

*Getting contradiction.* Indeed, the monodromy of the metric $f^*(g)$ (with respect to the developing map) is just the same as the monodromy of $f$, and I claim that the monodromy of a punctured surface with cusps at punctures and a finite number of conical points of angles $2\pi n$ can not lie in $A$.

Indeed, suppose by contradiction that the monodromy lies in $A$. Let us take the vector field $y\frac{\partial}{\partial y}$ on $\mathbb H$. Let $v=f^*(y\frac{\partial}{\partial y})$ be the pullback. Note that this pull-back field is well defined on $M\setminus \{crit( f)=conical \ points\}$ since $A$ preserves $y\frac{\partial}{\partial y}$ . Furthermore, this field $v$ has norm $1$ in the hyperbolic metric, it is analytic, and it is defined everywhere apart from the branching locus $crit (f)$. So the flow of this field is well defined outside of a subset of $M$ of measure zero. However, this field is contracting the area form corresponding to the metric (this is clear for the field $y\frac{\partial}{\partial y}$ on $\mathbb H$, in time $t$ it contracts the area form by the factor $e^{-t}$. The flow is an isometry on vertical lines and a contraction of horizontal lines, corresponding to horocircles of infinity on $\mathbb H$). At the same time, by Gauss-Bonnet formula, since $M$ has only finite number of cusps and finite number of conical points, the volume of $M$ with respect to the pull-back hyperbolic metric is finite. This is a contradiction.

Let us now prove that $f^*(g)$ is indeed a metric with finite number of cusps (close to the punctures of $M$) and finite number of conical points. Let $p$ be one of punctures of $M$ and let $\dot D\subset M$ be a punctured disk in $M$ whose puncture is at $p$. Then the monodromy around the puncture is an element $\rho\in A$. Clearly, the map $f$ induces to us a map $\tilde f: \dot D\to \mathbb H/\rho$. Now, the quotient $\mathbb H/\rho$ is either a) punctured disk or b) a cylinder, or c) a disk (if $\rho$ is the identity). In cases b) and c) the map $\tilde f$ extends to the whole disk $D$ and so in reality the monodromy around $p$ is trivial and we can extend $f$ to this puncture. If, on the other hand $\mathbb H/\rho$ is a punctured disk, again we can extend $\tilde f$ to the map from the whole $D$ to the one point completion of $\mathbb H/\rho$. In such a case $f^*(g)$ has a cusp at $p$. It is also not hard to see that the number of critical points of $f$ is finite, they can not accumulate towards the cusp. This finishes the proof of the statement.

Old answer. In this answer I was considering the trivial case when the monodromy is $\mathbb Z$.

If the monodromy of $f$ is $A=\left\{\begin{pmatrix} a & b \\ 0 & \frac{1}{a}\end{pmatrix}:a>0, b\in\mathbb{R}\right\},$ let $\mathbb Z$ denote the corresponding group acting on $\mathbb H$ and consider $\mathbb H/\mathbb Z$. This quotient is biholomorphic to either a punctured disk - in case $a=1, b\ne 0$, or a cylinder, if $a\ne 1$, or $\mathbb H^2$ if the monodromy is trivial. Clearly, from the multivalued map $f: M \to \mathbb H^2$ we get a genuine holomorphic map $\tilde f: M\to \mathbb H^2/\mathbb Z$.

Note finally that on a punctured disk, on a cylinder, or on $\mathbb H^2$, we can always construct a negative non-constant harmonic function $\phi$, and we can just take the pull-back $\tilde f^*(\phi)$ on $M$, which will also be negative, non-constant and harmonic on $M$.