This is detailed in the Azure online service terms: https://www.microsoftvolumelicensing.com/DocumentSearch.aspx?Mode=3&DocumentTypeId=46
At this moment these terms state the following about caching:
Customer may not cache or store information delivered by the Azure Maps API including but not limited to geocodes and reverse geocodes, map data tiles and route information (the “Results”) for the purpose of scaling such Results to serve multiple users, or to circumvent any functionality in Azure Maps.
Caching and storing Results is permitted where the purpose of caching is to reduce latency times of Customer’s application. Results may not be stored for longer than: (i) the validity period indicated in returned headers; or (ii) 6 months, whichever is the shorter. Notwithstanding the foregoing, Customer may retain continual access to geocodes as long as Customer maintains an active Azure account.
Customer may not display any Results, except geocodes solely as described in these Online Service Terms, on any third-party content or geographical map database.
My understanding (I'm not a lawyer, so can't give legal advice), if you are caching the data locally to do some calculations, you can cache route results up to 6 months and geocode results for as long as you have an Azure Maps account.