MySQL using BETWEEN comparison with NULL

IFNULL might help out here:

WHERE A.event_date BETWEEN IFNULL(B.start_date,"1900-01-01") 
                   AND IFNULL(B.end_date,now());

The COALESCE() function will return the first non-null value in the parameter list. Give this a try:

WHERE A.event_date BETWEEN COALESCE(B.start_date,'1900-01-01') and COALESCE(B.end_date,CURRENT_TIMESTAMP)