# The following two lines check your code and
# alert you to many common errors
use strict;
use warnings;

# Lets keep the filespec out the main code
# using my when defining a variable limits its scope. 
my $infile  = "rep_set_ass_tax.fna"; 
my $outfile = "seq_id.txt"; 

# No point going on if we can not open both files
# You were opening the outfile in a loop before, lots of opens
# and closes happening that were not needed
open my $in,  '<',  $infile  or die "Can't read $infile: $!\n"; 
open my $out, '>>', $outfile or die "Can't read $outfile: $!\n"; 

while (my $line=<$in>) { 
   if($line=~/^>/) {           # ^ added for only lines starting with > 
      my @vettore=split(/\s+/, $line);  
      foreach (@vettore) {    # can use for(@array) but they are the same
         print $out "$_ \n"; # Variables can be interpolated in a string
      }                       # end foreach loop	
   } 
} 
close $in;
close $out;