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 $M$.
