I can give you the answer for a certain class of Markov Chains, i.e. for *reversible*, *irreducible* and *aperiodic* Markov Chains on a finite state space.

Then the question is related to the spectral gap of the transition matrix $P:\Omega\times\Omega\to [0,1]$ of your chain. The transition matrix entry $P(x,y)$ is the probability of going from state $x$ to $y$. Thus one has $\sum_{y\in \Omega} P(x,y)=1$ 
If your chain is *irreducible* ($\forall x,y\in \Omega \exists n\in \mathbb{N}: P^n(x,y)>0$) and *aperiodic*. Aperiodic means that $\forall x\in \Omega$ the *period* defined by the greatest common divisor of the set $T(x)=\{n\in \mathbb{N}: P^n(x,x)>0\}$ is $1$. Then one has the following Theorem:

If $\Omega$ is finite and the chain defined by the transition matrix $P$ is irreducible and aperiodic than there exists a unique stationary distribution $\pi$ such that $\pi P =\pi$. 

A chain satisfying a *detailed balance* relation $\pi(x) P(x,y) =\pi(y) P(y,x)$ is *reversible*. 

Under these conditions the eigenvalues of $P$ are bounded in modulus by $1$ and the largest is $1$, the corresponding left eigenvector is $\pi$. Then the *spectral gap* is defined by $\gamma=1-\max_{\lambda\ne 1} |\lambda| $. Note that $0<\gamma<1$. Then it yields the following estimate for the variance of a test function $f:\Omega\to\mathbb{R}$ with respect to the stationary measure $\pi$

${\mathrm{Var}}_\pi(P^n f) \leq (1-\gamma)^{2t} {\mathrm{Var}}_\pi(f)$