Which Doctrine version are you using, because in Doctrine2 you should be using the QueryBuilder class and you should use class names and properties, not table names and fields in DQL.
Thus, you should join to the class field name, not the table field name.
innerJoin('t.request_details', 'r') // where request_details is a propery on Tester
Also, you do not need the where that joins them (where(t.id = r.tester_id)), this is managed by Doctrine and will work provided that the entities are properly mapped.
You also do not need the request_details r in the from part, Doctrine will take care of this too.
Also, use class names in from, not table names.
EDIT (forgot the getQuery() before getResults()):
In the end you query would look something like this:
$queryBuilder = EntityManager::create(//em options)->createQueryBuilder();
$queryBuilder->select('t.department')
->from('Tester', 't')
->innerJoin('t.request_details', 'r') // request details is a propery on Tester, that maps to RequestDetails
->where('r.request_id = ?1')
->setParameter(1, 123);
Doctrine will take care of turning this into SQL and joining the thing.
In the end you'll also need to fetch the stuff:
$departments = $queryBuilder->getQuery()->getResult();
EDIT:
for Doctrine 1, something like this should work:
$q = Doctrine_Query::create()
->select('t.department')
->from('Tester t')
->innerJoin('t.request_details r') // request details is a propery on Tester, that maps to RequestDetails
->where('r.request_id = ?', 123);
$depts = $q->fetchArray();
I am not really familiar with Doctrine 1, so take a look at this for more info:
http://docs.doctrine-project.org/projects/doctrine1/en/latest/en/manual/dql-doctrine-query-language.html