Long-run covariance matrix estimation is the building block of time series inference. The corresponding difference-based estimator, which avoids detrending, has attracted considerable interest due to its robustness to both smooth and abrupt structural breaks and its competitive finite sample performance. However, existing methods mainly focus on estimators for the univariate process while their direct and multivariate extensions for most linear models are asymptotically biased. We propose a novel difference-based and debiased long-run covariance matrix estimator for functional linear models with time-varying regression coefficients, allowing time series non-stationarity, long-range dependence, state-heteroscedasticity and their mixtures. We apply the new estimator to (i) the structural stability test, overcoming the notorious non-monotonic power phenomena caused by piecewise smooth alternatives for regression coefficients, and (ii) the nonparametric residual-based tests for long memory, improving the performance via the residual-free formula of the proposed estimator. The effectiveness of the proposed method is justified theoretically and demonstrated by superior performance in simulation studies, while its usefulness is elaborated via real data analysis. Our method is implemented in the R package mlrv.