I load it to ARGV to run it through a while( <> ) loop

This is the cause of your problems. Don't do this. If you want to operate on the individual lines use slurp lines and a for loop instead.

#!/usr/bin/env perl 
use strict;
use warnings;
use Path::Tiny;

###### Output dhcp configuration file
my $dhcpcdfile = 'd.conf';             # TEST

my @rows = path($dhcpcdfile)->lines;
my %ip_params = (); # YOU MUST POPULATE THIS FIRST
my $foundinterface; # YOU MUST ASSIGN A VALUE TO THIS

for my $line (@rows) {
     ### YOUR CODE HERE
     my @ip_fields = split( /=/,$line);
     # look for profile with matching interface name
     if ( $line =~ /^\s*profile\s+static_$ip_params{interface}\b.*\n/m
+ )
       {               # format matches 'profile static_eth0'
       }
     elsif( $foundinterface and $line =~ /^.*static\s*ip_address=/ ) {
       $line = "static ip_address=$ip_params{'ip_address'}\n";
       }
     elsif( $foundinterface ){
       last; # No need to continue looking through the file
       }
     ### YOUR CODE ENDS HERE
}

path('spew.cfg')->spew(@rows);
[download]

Note, untested because what was supplied doesn't compile and isn't an SSCCE.

Update: s/slurp/lines/;