No problem at all for DBI. I ran your SQL without problem in the postgres dialect:

#!/bin/env perl

use strict;
use warnings;
use Data::Dumper;
use DBI;

#  table my_table;
#  first | middle | last 
# -------+--------+------
#  joe   | john   | sean
#  joe   | john   | sean
#  joe   | john   | sean
#  pet   | joe    | sean
#  pet   | zoe    | joe
#  ken   | zoe    | joe
# (6 rows)

# (picking up dsn from environment:)
my $dbh = DBI->connect or die "oops - no db connection\n";
$dbh->{RaiseError} = 1;

my $sql = "
with data_count as (
    select
         sum(case when FIRST  = 'joe' then 1 else 0 end) as a_count
       , sum(case when MIDDLE = 'joe' then 1 else 0 end) as b_count
       , sum(case when LAST   = 'joe' then 1 else 0 end) as c_count
    from my_table
    where FIRST like 'joe%'
      or MIDDLE like 'joe%' or LAST like 'joe%'
)  
          select 'Search by: ' || 'joe'
union all select 'Found ' || cast(a_count as integer) || ' ' || 'joe' 
+|| ' for First'  from data_count
union all select 'Found ' || cast(b_count as integer) || ' ' || 'joe' 
+|| ' for Middle' from data_count
union all select 'Found ' || cast(c_count as integer) || ' ' || 'joe' 
+|| ' for Last'   from data_count
" ;

#         ?column?        
# ------------------------
#  Search by: joe
#  Found 3 joe for First
#  Found 1 joe for Middle
#  Found 2 joe for Last
# (4 rows)

my $sth = $dbh->prepare($sql);
   $sth->execute() or die "SQL Error: $DBI::errstr\n";

my $data = $sth->fetchall_arrayref({});

warn Dumper $data;
[download]

which results in the expected output (proving that neither DBI nor the general SQL is at fault):

$VAR1 = [
          {
            '?column?' => 'Search by: joe'
          },
          {
            '?column?' => 'Found 3 joe for First'
          },
          {
            '?column?' => 'Found 1 joe for Middle'
          },
          {
            '?column?' => 'Found 2 joe for Last'
          }
        ];
[download]

(A further variant with placeholders also gave no problem (in postgres). )

What database are you using? What is the error you get? Is there anything in the database logfile?