SELECT *
FROM mytable
WHERE (start_date, end_date) OVERLAPS ('2012-01-01'::DATE, '2012-04-12'::DATE);
As funções de data e hora são a seção relevante nos documentos.
SELECT *
FROM mytable
WHERE (start_date, end_date) OVERLAPS ('2012-01-01'::DATE, '2012-04-12'::DATE);