Edits: Changed function to eigenfunction. I should have stated the problem with more explicit conditions. Anyways I realized the original formulation is not true, even when one starts at a single state: take the function on the $\mathbb{Z}$ with $f(1) =1$, $f(-1) = -1$, $f(0)= 0$, and $f(x) = \text{sgn}(x) \epsilon$ where $\epsilon$ is very small, and the simple random walk on $\mathbb{Z}$, then it will have the highest variance at $t=1$. One can easily adapt this example to the simple random walk on the $n$-cycle.

Given a function $f: \Omega \to \mathbb{R}$, where $\Omega$ is the state space of an ergodic finite state Markov chain, and let the chain start at a single state $x \in \Omega$. Assume $f$ is an eigenfunction of the chain. Is it true that $\mathbb{E}_t (f- \mathbb{E}_t f)^2$ is nondecreasing in $t$? Here $\mathbb{E}_t f$ denotes $\mathbb{E} P_t f$ where $P_t$ is the Markov transition kernel from time $0$ to time $t$. Note it's important to start with the point mass distribution at a single state, since otherwise one could choose an initial distribution that has an $f$-variance larger than the stationary (as pointed out by one of the commenters below).

Wilson's method gives a way to bound the variance of eigenfunctions in $t$ in a way that's reminiscent of the Martingale difference method, but since the variance at time $\infty$ is usually easy to calculate, if we know the above monotonicity result, we could bound the variance at finite time easily.