It's easy to do date arithmetic in MySQL and other RDMS systems. You need all the records in which the start date is not after the year in question OR the end date is not before the year in question. That is this expression:
NOT(YEAR(start_date) > 2014 OR YEAR(start_date + INTERVAL period MONTH) < 2014)
This logically reduces to
YEAR(start_date) <= 2014 AND YEAR(start_date + INTERVAL period MONTH) >= 2014
So this query will do it.
SELECT whatever, whatever
FROM project
WHERE YEAR(start_date) <= 2014
AND YEAR(start_date + INTERVAL period MONTH) >= 2014
AND (whatever other selection criteria you have)
This will give all projects that were active during 2014, including those that started before 2014 and those that will still be in progress at the end of that year.