Perl 5 -p, 27 22 bytes

$_=!/[6-9].(?=(..)*$)/

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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.

Perl 5 -p, 27 22 18 bytes

Saved 4 bytes when @NahuelFouilleul pointed out that it doesn't need to be a look-ahead in the regex

$_=!/[6-9].(..)*$/

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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.

Perl 5 -MList::Util=max -p, 27 bytes

$_=60>max/..?(?=(?:..)*)$/g

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Since the input is guaranteed to be less than 236000, the hours can be ignored as they will always be valid.

The pattern match extracts all pairs of digits, accounting for the fact that the first might be only a single digit. Then, the resulting numbers are checked to ensure that the greatest of them is under 60.

Perl 5 -p, 27 22 bytes

$_=!/[6-9].(?=(..)*$)/

Try it online!

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.