It won't be the entire Shiny Server that's paused, but any sessions that are connected to that particular R process are going to have to wait. For Shiny Server Pro, you can use the utilization_scheduler directive in your shiny-server.conf file to specify how many or few browser sessions are allowed to connect to a single R process before SSP spawns a new R process. If you have any questions or issues getting this working, you can email support@rstudio.com.
In the next version of Shiny there will be support for asynchronous operations, which will give you finer grained control over this sort of thing, at the cost of making your application code more complicated (asynchronous logic has a different syntax than regular R code).