Explanation faithfulness of model predictions in natural language processing is typically evaluated on held-out data from the same temporal distribution as the training data (i.e. synchronous settings). While model performance often deteriorates due to temporal variation (i.e. temporal concept drift), it is currently unknown how explanation faithfulness is impacted when the time span of the target data is different from the data used to train the model (i.e. asynchronous settings). For this purpose, we examine the impact of temporal variation on model explanations extracted by eight feature attribution methods and three select-then-predict models across six text classification tasks. Our experiments show that (i) faithfulness is not consistent under temporal variations across feature attribution methods (e.g. it decreases or increases depending on the method), with an attention-based method demonstrating the most robust faithfulness scores across datasets; and (ii) select-then-predict models are mostly robust in asynchronous settings with only small degradation in predictive performance. Finally, feature attribution methods show conflicting behavior when used in FRESH (i.e. a select-and-predict model) and for measuring sufficiency/comprehensiveness (i.e. as post-hoc methods), suggesting that we need more robust metrics to evaluate post-hoc explanation faithfulness. Code will be made publicly available.