Perl 5 -p, 27 22 bytes
$_=!/[6-9].(?=(..)*$)/
Since the input is guaranteed to be less than 236000, the hours can be ignored as they will always be valid. This pattern match checks if there is a 6, 7, 8, or 9 in the tens digit of the minutes or seconds. The match is then negated to get truthy for valid dates and falsy for invalid ones.