A key summary statistic in a stationary functional time series is the long-run covariance function that measures serial dependence. It can be consistently estimated via a kernel sandwich estimator, which is the core of dynamic functional principal component regression for forecasting functional time series. To measure the uncertainty of the long-run covariance estimation, we consider sieve and functional autoregressive (FAR) bootstrap methods to generate pseudo-functional time series and study variability associated with the long-run covariance. The sieve bootstrap method is nonparametric (i.e., model-free), while the FAR bootstrap method is semi-parametric. The sieve bootstrap method relies on functional principal component analysis to decompose a functional time series into a set of estimated functional principal components and their associated scores. The scores can be bootstrapped via a vector autoregressive representation. The bootstrapped functional time series are obtained by multiplying the bootstrapped scores by the estimated functional principal components. The FAR bootstrap method relies on the FAR of order 1 to model the conditional mean of a functional time series, while residual functions can be bootstrapped via independent and identically distributed resampling. Through a series of Monte Carlo simulations, we evaluate and compare the finite-sample accuracy between the sieve and FAR bootstrap methods for quantifying the estimation uncertainty of the long-run covariance of a stationary functional time series.