The problem is that once the range exception is caught and the user is prompted again, the format exception can't be caught.

Encapsulate getting the user input ($start_date = <>; chomp ($start_date);) and checking its format into one routine (e.g. get_date()) that you then call every time you need new/corrected input.  This way, the format checking would also happen every time...

(Update) Something like this:

sub get_date {
    my $prompt = shift;
    while (1) {
        print "$prompt (YYYYMMDD): ";
        my $date = <>;
        chomp $date;
        if ($date !~ (/\d{8}/)) {
            print "ERROR: Invalid format!\n";
        } else {
            return $date;
        }
    }
}

my $today = ...

my $start_date;
while (1) {
    $start_date = get_date("Please enter the collection start date");
    if ($start_date < 18960101) {
        print "ERROR: Data prior to 18960101 is not available!\n";
    } elsif ($start_date > $today) {
        print "ERROR: Start date is in the future!\n";
    } else {
        last;
    }
}

# ditto for $end_date
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