Yet another approach... replace $test with whatever string you are looking for :) Good luck :)

use strict;

my (@BUFFER,@FINAL); 
my $num;
my $test = "Zwire6";
my $reorder = 0;
my $remove = 0;

while(<DATA>){

if(/$test/ig){ $remove = 1; } # set flag to remove if found

if(/\[server(\d+)\]/){
    if($remove == 1){ undef @BUFFER; $remove = 0; $reorder = 1;}
    if($reorder == 1) { $_ =~s/server$1/server$num/;}
        # renumber the servers 
    push @FINAL, @BUFFER;  # transfer BUFFER to FINAL
    undef @BUFFER;   # empty BUFFER 
    push @BUFFER, $_;  # start over :)
    $num = scalar $1;
    next;
    }
push @BUFFER, $_;
}

if($remove == 1){ undef @BUFFER; $remove = 0; $reorder=1; } 
push @FINAL, @BUFFER; #transfer the last


print @FINAL;   # or send it to yer handle :)


__DATA__
[reports]
yearlystore = 
quarterlystore = 
monthlystore = 
weeklystore = 
dailystore = 
[server1]
logfilepath1 = F:\Logfiles\ZWire\Zwire4\split-0\ex*.log
servername = Zwire4
logfilepassword = 
logfileusername = 
id = a104136320003954
[server2]
servername = Zwire6
logfilepath1 = f:\logfiles\zwire\zwire6\split-0\ex*.log
id = a104136346513171
logfilepassword = 
logfileusername = 
[server3]
id = a104136351461256
logfilepath1 = f:\logfiles\zwire\zwire7\split-0\ex*.log
logfilepassword = 
servername = zwire7
[download]

:)