Sorry the the slow reply here - I think the best really would be to follow @nirgrahamuk advice and start doing memory profiling of your shiny app. You either are using a package that creates a memory leak or you are for some reason allocating memory without ever deallocating it again.
In the metrics page of your app (e.g. https://www.shinyapps.io/admin/#/application/.../metrics) you should see the increase in memory use. Is this linear or step-wise ? Is it triggered by some specific operations ?
If nothing of the above helps, would you be able to create a reproducible example which we could analyse on our side ?
Also maybe take a look at Memory usage · Advanced R. where some possible ways to create memory leaks in code are referenced.