La combinación condicional de mysql depende de una columna

no puede crear uniones condicionalmente entre tablas. La mejor manera de hacerlo es usar LEFT JOIN y usar tu CASE en tu SELECT declaración. Debe especificar las columnas comunes que desea mostrar,

SELECT *,
        (CASE notification_base.type
            WHEN 'photo'
            THEN photo.columnName1
            ELSE post.ColumnName1
        END) as ColumnName1,
        (CASE notification_base.type
            WHEN 'photo'
            THEN photo.columnName2
            ELSE post.ColumnName2
        END) as ColumnName2
FROM notification_action
    INNER JOIN notification_base
        ON notification_action.not_base_id = notification_base.id
    INNER JOIN notification_sub
        ON notification_action.not_base_id = notification_sub.not_base_id
    INNER JOIN photo
        ON photo.id = notification_base.object_id
    INNER JOIN post
        ON post.id = notification_base.object_id
WHERE notification_sub.user_id = 3
    AND notification_sub.lastShowDate < notification_action.creationDate;