Merge rows if date columns are overlapping in TSQL

This approach uses an additional temp table to identify the groups of overlapping dates, and then performs a quick aggregate based on the groupings.

SELECT *, ROW_NUMBER() OVER (ORDER BY Id, Type) AS UID,
    ROW_NUMBER() OVER (ORDER BY Id, Type) AS GroupId INTO #G FROM #TempTable
WHILE @@ROWCOUNT <> 0 BEGIN
    UPDATE T1 SET
        GroupId = T2.GroupId
    FROM #G T1
        INNER JOIN (
            SELECT T1.UID, CASE WHEN T1.GroupId  < T2.GroupId THEN T1.GroupId ELSE T2.GroupId END
            FROM #G T1
                LEFT OUTER JOIN #G T2
                    ON T1.Id = T2.Id AND T1.Type = T2.Type AND T1.GroupId <> T2.GroupId
                        AND T1.StartDate <= T2.EndDate AND T2.StartDate <= T1.EndDate
            ) T2 (UID, GroupId)
            ON T1.UID = T2.UID
    WHERE T1.GroupId <> T2.GroupId
END
SELECT Id, MIN(StartDate) AS StartDate, MAX(EndDate) AS EndDate, Type
FROM #G G GROUP BY GroupId, Id, Type

This returns the expected values

Id          StartDate  EndDate    Type
----------- ---------- ---------- -----------
1           2012-02-18 2012-09-27 1
1           2014-08-23 2014-11-24 3
2           2015-07-04 2015-09-06 1
3           2013-11-01 2013-12-01 0
3           2018-01-09 2018-02-09 0

This is 2008 compatible. A CTE really is the best way to link up all overlapping records in my opinion. The date overlap logic came from this thread: SO Date Overlap

I added extra data that's more complex to make sure that it's working as expected.

DECLARE @Data table (Id INT, StartDate DATE, EndDate DATE, Type INT)
INSERT INTO @data 

SELECT 1,'2/18/2012' ,'3/18/2012', 1 UNION ALL
select 1,'3/17/2012','6/29/2012',1 UNION ALL
select 1,'6/27/2012','9/27/2012',1 UNION ALL
select 1,'8/23/2014','9/24/2014',3 UNION ALL
select 1,'9/23/2014','10/24/2014',3 UNION ALL
select 1,'10/23/2014','11/24/2014',3 UNION ALL
select 2,'7/4/2015','8/6/2015',1 UNION ALL
select 2,'8/4/2015','9/6/2015',1 UNION ALL
select 3,'11/1/2013','12/1/2013',0 UNION ALL
select 3,'1/9/2018','2/9/2018',0 UNION ALL 
select 4,'1/1/2018','1/2/2018',0 UNION ALL --many non overlapping dates
select 4,'1/4/2018','1/5/2018',0 UNION ALL
select 4,'1/7/2018','1/9/2018',0 UNION ALL
select 4,'1/11/2018','1/13/2018',0 UNION ALL

select 4,'2/7/2018','2/8/2018',0 UNION ALL --many overlapping dates
select 4,'2/8/2018','2/9/2018',0 UNION ALL 
select 4,'2/9/2018','2/10/2018',0 UNION all
select 4,'2/10/2018','2/11/2018',0 UNION all
select 4,'2/11/2018','2/12/2018',0 UNION all
select 4,'2/12/2018','2/13/2018',0 UNION all

select 4,'3/7/2018','3/8/2018',0 UNION ALL --many overlapping dates, second instance of id 4, type 0
select 4,'3/8/2018','3/9/2018',0 UNION ALL 
select 4,'3/9/2018','3/10/2018',0 UNION all
select 4,'3/10/2018','3/11/2018',0 UNION all
select 4,'3/11/2018','3/12/2018',0 UNION all
select 4,'3/12/2018','3/13/2018',0 



;
WITH cdata
AS (SELECT  Id,
        d.Type,
        d.StartDate,
        d.EndDate,
        CurrentStart = d.StartDate
    FROM    @Data d
    WHERE
        NOT EXISTS (
            SELECT * FROM @Data x WHERE x.StartDate < d.StartDate AND d.StartDate <= x.EndDate AND d.EndDate >= x.StartDate AND d.Id = x.Id AND d.Type = x.Type --get first records for overlapping ranges

                )       
    UNION ALL
    SELECT  d.Id,
        d.Type,
        StartDate = CASE WHEN d2.StartDate < d.StartDate THEN d2.StartDate ELSE d.StartDate END,
        EndDate = CASE WHEN d2.EndDate > d.EndDate THEN d2.EndDate ELSE d.EndDate END,
        CurrentStart = d2.StartDate
    FROM    cdata d
        INNER JOIN @Data d2
            ON (
                d.StartDate <= d2.EndDate
                AND d.EndDate >= d2.StartDate
            ) 
            AND d2.Id = d.Id
            AND d2.Type = d.Type
            AND d2.StartDate > d.CurrentStart)
SELECT cdata.Id, cdata.Type, cdata.StartDate, EndDate = MAX(cdata.EndDate) 
FROM        cdata 
GROUP BY cdata.Id, cdata.Type, cdata.StartDate