If you are using Perl 5.10 or newer then you can use the new given/when syntax to handle selecting between multiple code paths. However, anything of that nature is likely to be syntactic sugar (which may provide a programmer and maintainer efficiency gain) rather than an runtime improvement. No tinkering with code on that level is going to make any interesting runtime performance improvement however because any trivial change in runtime speed there will be completely hidden in I/O time.

A very minor speed improvement can be made by moving %monthNos population out of the loop. Turning off warnings for the entire scope is a bad idea. Either do it locally, or better still fix the error. With those changes and using given/when your code looks like:

#!/usr/bin/perl
use strict;
use warnings;
use 5.010;

my $no = 0;
my %monthNos =
    map {$_ => ++$no} qw{ Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov 
+Dec };

while (my $line = <DATA>) {
    chomp($line);
    my ($mon, $day, $time, $loghost, $prog, $remainder) =
        split m{:?\s+}, $line, 6;
    my ($user) = $remainder =~ m{user=([^,]+)};
    my ($rip)  = $remainder =~ m{rip=([^,]+)};
    my ($op)   = $remainder =~ m{\s(==|<=|\*\*|\+\+)\s};
    my $yr     = 2012;

    $remainder =~ tr/"/'/;
    $_ //= '' for $user, $rip;

    given ($op) {
        when ('==') {print "Do da ==\n";}
        when ('<=') {print "Doing <=\n";}
        when ('**') {print "Process **\n";}
        when (undef) {print "No op found\n";}
        default {print "Duh! Can't do $op\n";}
    }

    my $csv = sprintf q{%02d/%02d/%s %s,%s,%s,"%s",%s,%s},
        $day, $monthNos{$mon}, $yr, $time, $loghost, $prog, $remainder
+, $user,
        $rip;

    print "$csv\n";
}

__DATA__
May 2 07:06:20 l.net exim[1234]: 2012-05-02 07:06:20 1e <= inb@it.com 
May 2 07:06:20 l.net exim[1234]: 2012-05-02 07:06:20 1e <= inb@it.com 
May 2 07:06:20 l.net exim[1235]: 2012-05-02 07:06:20 1e => pp <pp@nsd.
+net> 
May 2 07:06:20 l.net exim[1235]: 2012-05-02 07:06:20 1e ++ pp <pp@nsd.
+net>
[download]

Prints:

Doing <=
02/05/2012 07:06:20,l.net,exim[1234],"2012-05-02 07:06:20 1e <= inb@it
+.com ",,
Doing <=
02/05/2012 07:06:20,l.net,exim[1234],"2012-05-02 07:06:20 1e <= inb@it
+.com ",,
No op found
02/05/2012 07:06:20,l.net,exim[1235],"2012-05-02 07:06:20 1e => pp <pp
+@nsd.net> ",,
Duh! Can't do ++
02/05/2012 07:06:20,l.net,exim[1235],"2012-05-02 07:06:20 1e ++ pp <pp
+@nsd.net> ",,
[download]