I assume you meant $\frac{1}{n}\sum_{t=1}^n f(X_t)$ converges to $\mathrm{E}_\pi (f)$. I'll also quibble and point out that the (classical) CLT doesn't give error rates in terms of n, but the Berry-Esseen theorem does.

In any case, to address your concrete question:

Are there similar theorems which give error rates in terms of n, the maximum value of f, and one (or several) of the frequently computed statistics of finite state Markov chains, like relaxation time, mixing time, covering time, etc?

There is a huge literature on such things. To get started, you could look at this recently published book by Levin, Peres, and Wilmer or

The answer seems to be yes: try this book in preparation by Aldous and Fillsurvey paper.

I assume you meant $\frac{1}{n}\sum_{t=1}^n f(X_t)$ converges to $\mathrm{E}_\pi (f)$. I'll also quibble and point out that the (classical) CLT doesn't give error rates in terms of n, but the Berry-Esseen theorem does.

In any case, to address your concrete question:

Are there similar theorems which give error rates in terms of n, the maximum value of f, and one (or several) of the frequently computed statistics of finite state Markov chains, like relaxation time, mixing time, covering time, etc?

There is a huge literature on such things. To get started, you could look at this recently published book by Levin, Peres, and Wilmer or this book in preparation by Aldous and Fill.