On multi-tenant scenarios I have seen the worst impact that RLS can have (filtering by tenantId/customerid). Execution plan of all existing queries will change, and you may see a good number of indexes will no longer be used since you are adding a new filtering. So, the performance impact you may see is slow queries or scans over tables (timeouts on queries that scan big tables) until you make adjustment to indexes.
My suggestion, you really might be better off to not use RLS and move to an Azure SQL elastic pool and horizontal sharding based on tenant iD or customer id.