Great suggestions have been given. Here's are some other observations/suggestions:

• Use a three-argument open.
• Use lexical file handles, e.g., my $fh, instead of FILE.
• Your first while reads all lines from the file, so there are none left for the second while
• Use lexical variables throughout.
• Your if(/\b$s\b/) matches against $_ when you've assigned a file line to $line.
• close the file when your done with it.

Give the above, consider the following:

use strict;
use warnings;

my ( @IPS, $IP );

print "Enter name:\n";
my $s = <>;
chomp $s;

open my $fh, '<', 'routers.txt' or die $!;
while ( my $line = <$fh> ) {

    if ( $line =~ /\b$s\b/ ) {
        my ($IP) = $line =~ /(\d+\.\d+\.\d+\.\d+)/;
        push( @IPS, $IP );
    }

}

close $fh;

print "$_\n" for @IPS;
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Hope this helps!