Calculating Feature Importance of Time Series Data
For time series data, 


Sensitivity analysis can help with overall Importance of a feature. For example, is  "Day of the week" a good feature for stock price forecasting. LIME is one approach that can help. Details :  https://arxiv.org/abs/1606.05386 . One simple way is to mask each feature and check the impact on model's performance. 
Auto-corelation and Seasonality removal  (Details in tutorial at end of the answer) 
SHAP : (SHapley Additive exPlanations) is good at identifying features that impact output with lag (https://medium.com/datadriveninvestor/time-step-wise-feature-importance-in-deep-learning-using-shap-e1c46a655455)


End to end example : 
https://machinelearningmastery.com/feature-selection-time-series-forecasting-python/