or ...

my $sql = qq|
    SELECT ID, tutorID, Latitude, Longitude, City, State, Zipcode
      FROM $table_tzips
     WHERE Zipcode IS not NULL
     ORDER BY tutorID
|;

my $last_tutor_id = "";
my $last_dist;
my @tutors;
my $sth = $dbh->prepare($sql);
$sth->execute();
while (my $tzips = $sth->fetchrow_hashref()) {
    my $tutor_id = $tzips->{tutorID};
    my $dist = calculate_distance(
        $clong, $clat,
        $tzips->{Longitude}, $tzips->{Latitude} );

    if ($last_tutor_id ne $tutor_id) {
        $last_tutor_id = $tutor_id;
        push @tutors, $tzips;
    }
    else {
        next if $last_dist <= $dist;
        $tutors[-1] = $tzips;
    }
    $tzips->{Dist} = $last_dist = $dist;
}
my @report_fields = qw( tzipID ID Dist City State Zipcode );

for my $tutor ( sort{ $a->{Dist} <=> $b{Dist} } @tutors) ) {
    print(join(', ', @{$tutor}{@report_fields}), "\n");
}
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In the OP case, where the full data has to be keep in memory for the final sorting, this approach may be just more complicated than ikegami's one and also imposes an extra load in the database.

But when you can output the data as you go and don't need to keep it in memory for a final processing stage, it may perform better, specially for large data sets, as it's memory usage is fixed O(1), not dependant on the data set size O(N).