How to avoid Functions from a Performance Stand-point in Azure SQL Database?
You can rewrite it as an inline TVF returning a single column and row and CROSS APPLY it to get the benefits of inlining now (parallelism, no overhead of switching execution contexts, holistic query costing and optimisation) without having to wait for the work done on inlining of Scalar UDFs to get released.

So your function definition would be 

CREATE FUNCTION [dbo].[fnGetLocalDate] (@DateToConvert DATETIMEOFFSET = NULL) 
RETURNS TABLE 
AS 
    RETURN 
      SELECT CONVERT(DATETIMEOFFSET, @DateToConvert AT TIME ZONE 
                                     (SELECT Value 
                                      FROM   LookUp.Config 
                                      WHERE  Property = 'TimeZone')) AS 
                    LocalDate 


With example usage

SELECT o.name, L.LocalDate
FROM sys.objects o
CROSS APPLY [dbo].[fnGetLocalDate](o.modify_date) AS L


You don't really need the CASE as it will return NULL on NULL input anyway and may get better plans without it