$net="192.168.4.";
$counter=10;
...
$ip=$net.$counter;
...
while ( $counter <= 14 ) {
        ...
        open(LOGF,">>",$logpath . $ip);
        ...
                while ( <INFILE> ) {
                        ...
                        $ip=$net.$counter;
                        ...
                        }
                }
        $counter++;
        }
[download]

That would probably be better as:

my $net = "192.168.4.";
...
for my $counter ( 10 .. 14 ) {
        my $ip = $net . $counter;
        ...
        open(LOGF,">>",$logpath . $ip);
        ...
                while ( <INFILE> ) {
                        ...
                        }
                }
        }
[download]

                $|++;
[download]

At this point in the program the currently selected filehandle is STDOUT so that is the filehandle that will be affected by the change in $|.    In other words, LOGF will not be affected.

                        @array = split(" ",);
[download]

That is the same as saying:

                        @array = split;
[download]

#Only grab things associated with the current searched for ip and ends
+ with .com .net and so on

                        if (($array[7] =~ /[com|org|net|gov|cc|mil]\/$
+/)&&($array[3] eq $ip)) {
[download]

In your regular expression the character class [com|org|net|gov|cc|mil] will match one character only and is exactly the same as using [cegilmnortv|], and you have nothing matching the period character like in .com or .net.