Solution Overview
I made 2 Experiments on that issue, the first experiments showed that in case of stored procedures with no parameters, nothing changed in SQL Server 2016 and SSIS 2016, the first Result Set is returned and others are ignored.
The second experiment showed that when using parameters, this will throw an exception, so you have to define metadata using WITH RESULT SETS option, then remove this option.
Detailed Solution
Experiment 1
The following experiment are made using SQL Server 2016 and Visual Studio 2015 with SSDT 2016
- First i created this stored procedure - CREATE PROCEDURE sp_Test
AS
BEGIN
SET NOCOUNT ON;
SELECT TOP 10 PersonType,NameStyle,Title 
  FROM [AdventureWorks2016CTP3].[Person].[Person]
SELECT  TOP 10 PersonType,Firstname,Lastname
  FROM [AdventureWorks2016CTP3].[Person].[Person_json]
END
GO
 
- Then i added a Data flow task to SSIS package
- Added an OLEDB Source, Recordset destination
- In OLEDB source i select the Data access mode to SQL command
- an use the following commnad - EXEC sp_Test
 

- When clicking on Columns Tab it shows the first ResultSet structure

- And we i executed the package it runs succesfully

Experiment 2
I changed the stored procedures to the following:
ALTER PROCEDURE [dbo].[sp_Test]
    @param1 varchar(10),
    @param2 varchar(10),
    @param3 varchar(10)
AS
BEGIN
    SET NOCOUNT ON;
    SELECT TOP 10 PersonType,NameStyle,Title ,@param2 as 'Param'
  FROM [AdventureWorks2016CTP3].[Person].[Person]
    SELECT  TOP 10 PersonType,Firstname,Lastname,@param3 as 'Param'
  FROM [AdventureWorks2016CTP3].[Person].[Person_json]
END
And i used the following SQL Command in the OLEDB Source:
EXEC sp_Test ?,?,?
WITH RESULT SETS (
(
    PersonType NVarchar(10),
    NameStyle NVarchar(10),
    Title  NVarchar(10),
    Param Varchar(10)
)
)
And i mapped the parameters correctly.


When running the package it throws the following exception.
[OLE DB Source 2] Error: SSIS Error Code DTS_E_OLEDBERROR.  An OLE DB error has occurred. Error code: 0x80040E14.
  An OLE DB record is available.  Source: "Microsoft SQL Server Native Client 11.0"  Hresult: 0x80040E14  Description: "EXECUTE statement failed because its WITH RESULT SETS clause specified 1 result set(s), and the statement tried to send more result sets than this.".
After that i tried to remove the With RESULT SETS option, so the command is :
EXEC sp_Test ?,?,?
I tried to execute the package again, so it is executed with no errors.
Conclusion
Try to use the WITH RESULT SETs option to define the OLEDB Source metadata, after that the metadata is defined, just remove this option and run the package, so it will just take the first Result Set succesfully.