Alternatively to japhy's suggestion, move the declaration for @lines up above the while loop, and push the values of $lname, $fname, etc, onto the array during each iteration of the loop. As you have it written at present, @lines contains only the last set of values that was entered. So:

my @lines;
my $quit = 0;
...
until ($quit ) {
  ...
  push( @lines, ... );
  $quit = ( $ans =~ /^n/i );
}
foreach (@lines) {
  ...
  write ADDRESS;
}
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UPDATE: Another alternative, which would make this sort of program a lot easier to use: some Tk widgets. E.g. a pair of radio buttons for business vs. personal, four entry boxes for the names and address stuff, a "Store it" button, an "Exit" button... heck, why not a "Show list" button with a pop-up text widget, so the user can see who's on the list already. More features will come to mind (this can become addictive). You do have the Tk modules, don't you?