Using php, how to insert text without overwriting to the beginning of a text file

I'm not entirely sure of your question - do you want to write data and not have it over-write the beginning of an existing file, or write new data to the start of an existing file, keeping the existing content after it?

To insert text without over-writing the beginning of the file, you'll have to open it for appending (a+ rather than r+)

$file=fopen(date("Y-m-d").".txt","a+") or exit("Unable to open file!");

if ($_POST["lastname"] <> "")
{
   fwrite($file,$_POST["lastname"]."\n");
}

fclose($file);

If you're trying to write to the start of the file, you'll have to read in the file contents (see file_get_contents) first, then write your new string followed by file contents to the output file.

$old_content = file_get_contents($file);
fwrite($file, $new_content."\n".$old_content);

The above approach will work with small files, but you may run into memory limits trying to read a large file in using file_get_conents. In this case, consider using rewind($file), which sets the file position indicator for handle to the beginning of the file stream. Note when using rewind(), not to open the file with the a (or a+) options, as:

If you have opened the file in append ("a" or "a+") mode, any data you write to the file will always be appended, regardless of the file position.