What is the most efficient method of performing the FIZZBUZZ test in SQL Server?

Using a SQL Server 2014 memory optimized table and a natively compiled procedure:

-- Setup
CREATE DATABASE InMem;
GO
ALTER DATABASE InMem
ADD FILEGROUP FG1
CONTAINS MEMORY_OPTIMIZED_DATA;
GO
ALTER DATABASE InMem
ADD FILE 
(
    NAME = 'FN1', 
    -- Change to suit your system
    FILENAME = 'C:\Program Files\Microsoft SQL Server\MSSQL12.SQL2014\MSSQL\DATA\FN1.mod'
)
TO FILEGROUP FG1;
GO
USE InMem;
GO
CREATE TYPE dbo.FizzBuzzTableType AS TABLE 
(
    n integer NOT NULL INDEX i,
    FizzBuzz varchar(8) NOT NULL
) WITH (MEMORY_OPTIMIZED = ON);
GO

Native procedure:

CREATE PROCEDURE dbo.FizzBuzz
WITH 
    NATIVE_COMPILATION, 
    SCHEMABINDING, 
    EXECUTE AS OWNER
AS
BEGIN ATOMIC 
WITH 
(
    TRANSACTION ISOLATION LEVEL = SNAPSHOT, 
    LANGUAGE = N'english'
)   
    DECLARE @n AS dbo.FizzBuzzTableType;

    DECLARE @i integer = 1;
    WHILE @i < 50
    BEGIN
        IF @i % 15 = 0
        BEGIN
            INSERT @n (n, FizzBuzz) 
            VALUES (@i, 'FizzBuzz')
        END
        ELSE 
        BEGIN
            IF @i % 3 = 0
            BEGIN
                INSERT @n (n, FizzBuzz)
                VALUES (@i, 'Fizz')
            END
            ELSE 
            BEGIN
                IF @i % 5 = 0
                BEGIN
                    INSERT @n (n, FizzBuzz) 
                    VALUES (@i, 'Buzz')
                END
                ELSE
                BEGIN
                    INSERT @n (n, FizzBuzz) 
                    VALUES (@i, CONVERT(varchar(8), @i));
                END;
            END;
        END;

        SET @i += 1;
    END;

    SELECT
        N.n, 
        N.FizzBuzz
    FROM @n AS N
    ORDER BY
        N.n;
END;

Test:

SET NOCOUNT ON;
PRINT SYSUTCDATETIME();
GO
DECLARE @T AS dbo.FizzBuzzTableType;

INSERT @T (n, FizzBuzz)
EXECUTE dbo.FizzBuzz;
GO 100

PRINT SYSUTCDATETIME();

Typical results:

-- 95ms for 100 iterations, < 1ms each
2014-12-31 10:07:13.7993355
Beginning execution loop
Batch execution completed 100 times.
2014-12-31 10:07:13.8943409

This writes the procedure output to an in-memory table variable, because otherwise we're just testing the speed of displaying results in SSMS.

One million rows

The above native procedure takes about 12 seconds to run on 1,000,000 numbers. There are all sorts of faster ways to do the same thing in T-SQL. One I have written before follows. It runs in about 500ms on my laptop on a million rows when the intended parallel plan is achieved:

IF  OBJECT_ID(N'tempdb..#Result', N'U') IS NOT NULL
    DROP TABLE #Result;

IF  OBJECT_ID(N'tempdb..#Thousand', N'U') IS NOT NULL
    DROP TABLE #Thousand;

SET NOCOUNT ON;
DECLARE @start datetime2(7) = SYSUTCDATETIME();

CREATE TABLE #Thousand 
(
    n integer NOT NULL,

    CONSTRAINT PK_#Thousand
    PRIMARY KEY CLUSTERED (n)
);

-- Add 1,000 rows numbered 0-999 to #Thousand
WITH 
    L1 (n) AS
(
    SELECT  V.n
    FROM    
    (
        VALUES  (0), (1), (2), (3), (4),
                (5), (6), (7), (8), (9)
    ) AS V (n)
),
    Thousand AS
(
    SELECT  n = 
        CONVERT
        (
            integer,
            ROW_NUMBER() OVER (
            ORDER BY (SELECT NULL))
            - 1
        )
    FROM L1
    CROSS JOIN L1 AS L2
    CROSS JOIN L1 AS L3
)
INSERT #Thousand (n)
SELECT n
FROM Thousand;

-- To hold the Fizz Buzz output
CREATE TABLE #Result 
(
    n integer NOT NULL, 
    result varchar(8) NOT NULL
);

INSERT #Result
SELECT 
    Million.n, 
    Million.result
FROM
(
    -- Modulo operation to encourage few outer rows parallelism
    SELECT  n
    FROM    #Thousand
    WHERE   n % 1 = 0
) AS T1
-- Outer Apply to keep the Compute Scalar parallel
OUTER APPLY
(
    SELECT
        F2.n, 
        F2.result
    FROM #Thousand AS T2
    CROSS APPLY
    (
        -- Row numbers 1 to 1,000,000
        SELECT  (T1.n * 1000) + T2.n + 1
    ) AS F1 (n)
    CROSS APPLY
    (
        -- The Fizz Buzz bit
        SELECT
            F1.n,
            result =
                CASE 
                    WHEN F1.n % 15 = 0 THEN 'FizzBuzz'
                    WHEN F1.n % 3 = 0 THEN 'Buzz'
                    WHEN F1.n % 5 = 0 THEN 'Fizz'
                    ELSE CONVERT(varchar(8), F1.n)
                END
    ) AS F2
) AS Million
OPTION  (MAXDOP 4, QUERYTRACEON 9481);

PRINT DATEDIFF(MILLISECOND, @start, SYSUTCDATETIME());

This one runs the same on my machine as your first one (0ms). I'm not sure if it would scale faster or not.

;WITH t AS (
    SELECT RowNum = ROW_NUMBER() OVER (ORDER BY o.object_id)
    FROM sys.objects o
)
SELECT t.RowNum
    , Cxa.Fizz + CxB.Buzz
FROM t
CROSS APPLY (SELECT CASE WHEN t.RowNum % 3 = 0 THEN 'FIZZ' ELSE '' END) CxA(Fizz)
CROSS APPLY (SELECT CASE WHEN t.RowNum % 5 = 0 THEN 'BUZZ' ELSE '' END) CxB(Buzz)
WHERE t.RowNum < 50;

The best version I came up with runs in 30ms on my machine:

WITH t AS (
    SELECT 1 As RowNum
    Union ALL
    Select RowNum + 1
    From t
    Where RowNum < 49
)
SELECT t.RowNum
, SubString('FIZZ', (t.RowNum % 3)*10, 5) + SubString('BUZZ', (t.RowNum % 5)*10, 5)
FROM t;