That depends. I have some experience with BI solution (for example, we worked with Tableau), and it can operate is two main modes: It can execute the query against your server, or can collect the relevant data and store it on the user's machine (or on the server where the app installed). When working with large volumes, we used to make Tableau query the SQL Server itself, that's because our SQL Server machine is very strong compared to the other machines we had.
In any way, even if you store the data locally and want to "refresh" it, when it updates the data it needs to retrieve it from the database, which sometimes can also be an expensive operation (depends on how your data is built and organized).
You should also notice that you compare 2 different families of products: while Google BigQuery and Amazon's RedShift are actually database engines that used to store the data and also query it, most of the BI and reporting solutions are more concerend about querying the data and visualizing it and therefore (generally speaking) are less focused on having smart internal databases (at least from my experience).