Let $(X_n)_{n\in\mathbb{N}_0}$ be a Markov chain with state space $E$ and transition matrix $P=(p_{i,j})_{i,j\in E}$.
A real valued function $h$ on $E$ is called superharmonic if $h(x)\geq Ph(x)$ for all $x\in E$, where $Ph(x)=\sum_{y\in E}p_{x,y}h(y)$. Define the space of superharmonic functions by $$ \mathcal{S}(X,P):=\left\{h\colon E\to\mathbb{R}: h\geq Ph\right\}. $$
Show by way of example that the space of superharmonic functions $\mathcal{S}(X,P)$ is no linear space.
Edit
Consider a superharmonic function $h\in\mathcal{S}(X,P)$ with $h(x)>Ph(x)~\forall x\in E$.
Set $g(x):=-h(x)~\forall x\in E$. Then for any $x\in E$ it is $$ g(x)=-h(x)<-Ph(x)=P(-h(x))=Pg(x). $$ Thus $g\notin\mathcal{S}(X,P)$, hence $\mathcal{S}(X,P)$ is not a linear space, because in order to be a linear space it has to be $\alpha\cdot h\in\mathcal{S}(X,P)~\forall\alpha\in\mathbb{R}$. But this is not the case for $\alpha=-1$ as shown.
Am I right?
Your approach is certainly right: superharmonic functions form a convex cone, that is if $f$ and $g$ are superharmonic, then $\alpha f + \beta g$ is superharmonic for non-negative $\alpha$ and $\beta$. Moreover, if $f$ and $-f$ are superharmonic then $f$ is harmonic, so the whole question is to construct an example when there are superharmonic functions that are not harmonic. My guess is that example must be concrete: you have to provide a particular Markov Chain and a particular function. Can you think of such a particular example?
Example: since a single-state MC is not enough, let's go for two-state one: X = $\{0,1\}$. Put all the transitions to $\{0\}$ so that $Pf(x) = f(0)$. Hence harmonic functions must be constant, and any other function is either superharmonic, or subharmonic.