Hi, let me comment your code and see what exactly do you want to do:

#!/usr/bin/perl -w    # "-w" and "use warnings" are equivalents, it's prefered to use the second
    use warnings;
    use strict;  
    use 5.010; # You don't need back-compatibility, you can remove this

my $comp = $ARGV[0];      
# opening files are OK                            
if ( ! open ( FILE_HANDLE , "<" , $comp ) ) {      
        die "Error can't find the file: $comp because $!";}
if ( ! open ( OUT_HANDLE , ">" , $comp . ".comp.csv" ) ) { 
        die "Error can't make new file file: ${$comp}.comp.csv because $!";}

my @arr = ();
my $gene;  

# here is where you are iterating over the file, reading line by line
while (my $lineContent = <FILE_HANDLE> ) {
   chomp $lineContent;
   if ( $lineContent =~ m /^\#/){  # Skip comments
     next;
   }elsif($lineContent =~ m /\.raxtree$/){ # Skip lines terminated with "raxtree"
    next;
   }elsif($lineContent =~ m /^WARNING/){ # Skip lines that starts with "WARNING"
    next;
   }elsif($lineContent =~ m /^f/){ # Skip lines that start with "f"?
    next;
   }else{
   print "$lineContent\n"; # print lines that are not in previous patterns, why?
 }
# Here you are out of your file because the while {} loop ended.
foreach my $line ($lineContent){
   # I guess you want to parse this line and get the gene name, saving in a variable and then check lines with that gene, saving in a array
   if($line =~ m /^RAxML\_result\.(\w+)\.aa\.fa\.aln\.phy\.raxtree\.test\.\d/){
      $gene="$1\n";
           if($line =~ m /^(\w+)\s/){
       push @arr,$1;
     }
   }
 }
}
print $gene;
print @arr;

Guessing I can suggest:

#!/usr/bin/perl

use warnings;
use strict;

my $comp = $ARGV[0];
if ( ! open (FILE, "<" , $comp) ) {
        die "Error can't find the file: $comp because $!";}
if ( ! open (OUT, ">" , $comp . ".comp.csv") ) {
        die "Error can't make new file file: ${$comp}.comp.csv because $!";}

my %speciesByGene = ();
my $gene;

while (my $line = <FILE>) {
   chomp $line;
   if($line =~ m/^RAxML_result\.(.+)\.aa\.fa\.aln\.phy\.raxtree\.test\.\d+/){
       $gene = $1;
   }
   elsif ($line =~ m/^\#/){
       next;
   }elsif($line =~ m/\.raxtree$/){
       next;
   }elsif($line =~ m/^WARNING/){
       next;
   }elsif($line =~ m/^f/){
       next;
   }else{
       $line =~ s/\s.*//; # keeps species
       push (@{ $speciesByGene{$gene} }, $line);
   }
}
close FILE;

# print results
foreach $gene (sort keys %speciesByGene) {
    print OUT "$gene: ";
    print OUT join ",", @{ $speciesByGene{$gene} };
    print OUT "\n";
}
close OUT;

Output:

EOG09371BVG: IT_ine1,IM_nor1,OH_azt1,IE_sup1,IT_ras1,OD_mag1,OE_aff1