The main issue is that you are creating an array of arrays, which you then need to dereference: 
use warnings;
use strict;

my @retCodes_values;
while (<DATA>) {
    next unless /retCode\=/;
    chomp;
    my ( $string, $exit_code ) = split /retCode\=/, $_, 2;
    push @retCodes_values, [ $string, $exit_code ];
}

for my $aref (@retCodes_values) {
    if ($aref->[1] != 3) {
        print "@{ $aref }\n";
    }
}

__DATA__
15288 :: Mon Mar 2 13:23:32 2015::App.pm:status returned with retCode=
+1
15288 :: Mon Mar 2 13:24:32 2015::App.pm:status returned with retCode=
+3
15288 :: Mon Mar 2 13:25:32 2015::App.pm:status returned with retCode=
+4
15288 :: Mon Mar 2 13:23:32 2015::App.pm:status returned with retCode=
+5

[download]

Outputs: 

15288 :: Mon Mar 2 13:23:32 2015::App.pm:status returned with  1
15288 :: Mon Mar 2 13:25:32 2015::App.pm:status returned with  4
15288 :: Mon Mar 2 13:23:32 2015::App.pm:status returned with  5

[download]

See also:

In reply to Re^3: Do not run code even if one element has matched exit code value by toolic
in thread Do not run code even if one element has matched exit code value by jayu_rao

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