On top of what hippo says, you are reading and parsing the mapping files anew for every line you process, that's an awful lot of wasted work. Similarly, you are splitting the line into @source repeatedly for each mapping record.

As a first step: separate out the reading and parsing of the mapping files into data structures, do that once, then walk through the data structures in the loop over @lines. That might look something like this:

# Mapping des charges directes
my @mapping1 = map {
  my @mapping = split /\t/, $_;
  # account, mapped account, mapped source
  [ $mapping[0], $mapping[2], $mapping[3] ];
} <MAPPINGFILE1>;

# Mapping des charges indirectes
my @mapping2 = map {
  my @mapping = split /\t/, $_;
  # account, mapped account, mapped source
  [ $mapping[0], $mapping[4], $mapping[2] ];
} <MAPPINGFILE2>;


LINE: for my $line (@lines) {
  my $source = (split /\t/, $line)[2];

  # Mapping des charges directes
  for my $mapping1 (@mapping1) {
    my($account, $mapped_account, $mapped_source) = @$mapping1;

    # Account is matching with source
    if ($line =~ /$account/) {
      # Account substitution
      if ($mapping eq "") {
        $line =~ s/$account/"Compte cible non défini !"/;
      } else {
        $line =~ s/$account/$mapped_account/;
      }
      # Mapping = target Unit, Alloc_ + Unit source
      if ($mapped_source eq 'Unit source') {
        # Mapping : source Unit, Alloc_ + source Unit
        $line =~ s/$source/$source\tALLOC_$source/;
      } elsif ($mapped_source eq "") {
        $line =~ s/$source/"Unit cible non définie !"\tALLOC_$source/;
      } else {
        # Unit substitution
        $line =~ s/$source/$mapped_source\tALLOC_$source/;
      }
      push @lines2, $line;
      next LINE;
    }
  }

  # Mapping des charges indirectes
  for my $mapping2 (@mapping2) {
    my($account, $mapped_account, $mapped_source) = @$mapping2;
    if ($line =~ /$account/) {
      $line =~ s/$account/$mapped_account/;
      $line =~ s/$source/$mapped_source\tALLOC_$source/;
      push @lines2, $line;
      next LINE;
    }
  }
  push @rejects, "Lignes non mappées (Account): \t$line";
}
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However I suspect that even bigger savings are possible: for example, if the account name appears in a specific column in $line, you could probably turn the whole thing into a single hash lookup.