Save the IP of the visitor in a text file, but how can I not save it again if it is already there and how to read it?

As people have already commented in the comments, you'd better use something other than IP to mark the vote, but if you still want to check if an IP already voted, use something like:

$arquivo = "ips.txt";

function has_voted($ip) {
  // o método file retorna um array com cada linha do arquivo
  $rows = file($arquivo);

  // basta verificar se o ip do usuário já está gravado no arquivo.
  in_array($ip, $rows);
}

if (!has_voted($_SERVER['REMOTE_ADDR'])) {
  vota(); // seu método para votar

  // grava o ip do cara que acabou de votar no arquivo de ips
  // http://pt.stackoverflow.com/a/31970/2321
  $ip = $_SERVER['REMOTE_ADDR']  . "\n";
  file_put_contents($arquivo, $ip, FILE_APPEND | LOCK_EX);
}

Some considerations:

a) Save your file as json, because you can always use it with other languages. It might look something like this:

{"ipss":["111.111.111.111","222.222.222.222"]}

and save as file_name.json. You can also use comma-separated values (CSV), but I prefer json, which I find more descriptive.

b) The more ips you have, the slower it will be and the execution of the function, because where we are going to act is a collection, then as the collection increases, more data we have to act. The databases are optimized for this.

c) I did not include opening and saving the file, as you already know how to do it.

function check_ip($novo_ip)
{
    $sip = NULL;

    //aqui abre o arquivo json
    $ips = '{"ipss":["111.111.111.111","222.222.222.222"]}';

    //verifica se o ip é válido
   if (!filter_var($novo_ip, FILTER_VALIDATE_IP)) 
        return "Não pode votar";

    //passa json para array
    $o = json_decode($ips,TRUE);

    //já votou, não pode votar mais
    if(is_array($o['ipss']) && in_array($novo_ip, $o['ipss']))
        return "Não pode votar";

     $o['ipss'][]="$novo_ip";

     //cria miolo do json
     foreach ($o['ipss'] as $value) 
     {

            $sip .='"'.$value."',";
    }

    //aqui eu coloquei return mas pode colocar para gravar o aquivo atualizado
    return '{"ipss":['.trim($sip,",").']}';

}