Hyperparameter Tuning Time Series in Production
Some ideas:


Number of previous observations to use: depends on the process you are modelling. If the target is related in some way to many of the previous values, you may need to use more data in your tuning process. On the other hand, if the target is largely unrelated to previous observations you can use fewer. You will need to check the relationship - lag plots and auto-correlation plots may be useful.
Frequency of parameter tuning: again, depends on your model and process. It lay be a good idea to monitor the prediction accuracy of your model, and to only perform more parameter tuning if the model accuracy drops below a certain threshold.