Case Statement With Group By
Untested, but I believe your query should look like:

SELECT salesman
      ,sum(case when DATEPART(Year, warrantypurchasedate) = '2015'
                 and DATEPART(Quarter, warrantypurchasedate) = '1'
                then itemprice 
           end) As [Q1 2015]
      ,sum(case when DATEPART(Year, warrantypurchasedate) = '2016'
                 and DATEPART(Quarter, warrantypurchasedate) = '1'
                then itemprice 
           end) As [Q1 2016]
FROM Barnacles
WHERE DATEPART(Year, warrantypurchasedate) in ('2015', '2016')
  AND DATEPART(Quarter, warrantypurchasedate) = '1' 
GROUP BY salesman


I moved  sum outside of case, and added a WHERE clause to filter out relevent rows. I also removed date from the GROUP BY clause, otherwise you will get 1 row per date.