because you are placing the text at the same position

$txt->translate( 100, 650 );
$txt->text( $header, -encoding => 'utf8' );

$txt->translate( 100, 650 );
$txt->text( $quote, -encoding => 'utf8' );
[download]

Use your variable $linepos for all the elements and decrement to move down the page as required. To wrap the text you can use the paragraph method.

Thanks so much for the cleaner treatment. I didn't understand that the numbers in this context are supposed to get smaller as you go down the page. And it does not give you another page automatically: you have to ask for it, but this is where I have to stop for now.

Output:

C:\cygwin64\home\Fred\pages2\list>perl list11.pl
Wed, May 25, 2016
28. Don't be afraid to give up the good to go for the great.
Don't be afraid to give up the good to go for the great.
tasks for second page are fold sail
new_name is May_25_2016.pdf
[download]

#!/usr/bin/perl
use v5.14;
use utf8;
use PDF::API2;
use Time::Piece;
use Time::Seconds;

my $t = Time::Piece->localtime + ONE_DAY;
my $today = $t->strftime("%a, %b %d, %Y");
print "$today\n";
my $header = "List for $today";

my $pdf = PDF::API2->new(
   width  => 595,    # A4 dimensions in point
   height => 842,    # 1 point = 1/72 inch
);

my $page = $pdf->page;
my $txt  = $page->text;
my $font = $pdf->corefont('Times-Roman');

#header
my $linepos = 650;
$txt->font( $font, 32 );
$txt->translate( 100, $linepos );
$txt->text( $header );
$linepos -= 50;

#add randomtext
my $quote = get_quote();
print "$quote\n";
$txt->font( $font, 20 );
$txt->lead(20); # line spacing
$txt->translate( 100, $linepos );
$txt->paragraph($quote, 400, 100, -align => "left" );
my $overflow_text = $txt->textpos();
$linepos -= 100;

my @task = (
   'wake no later than 8',
   'stretch/take meds',
   'make coffee/eat cereal',
   'eat salad', 'bring food',
      'make coconut phone',
   'check provisions',
   'tie knots', 'fold sail',,
);

$txt->font( $font, 20 );

my $vspace = 70;
while ( (@task)&&($linepos>0) ) {
  my $item = shift @task;
  my $msg = '[  ]  ' . $item;
  drawline( $page, $linepos );
  $txt->translate( 70, $linepos+10 );
  $txt->text($msg);
  $linepos -= $vspace
}

say "tasks for second page are @task";

my $new_name = $t->strftime("%b_%d_%Y.pdf");
say "new_name is $new_name";
$pdf->saveas($new_name);

sub drawline {
  my ( $page, $y ) = @_;
  my $x1 = 50;
  my $x2 = 550;
  my $line = $page->gfx;
  $line->linewidth(3);
  $line->move( $x1, $y );
  $line->line( $x2, $y );
  $line->stroke;
}

sub get_quote {
   use HTML::TreeBuilder 5 -weak;

   my $site = 'http://motivationgrid.com/50-inspirational-quotes-to-li
+ve-by/';
   my $tree = HTML::TreeBuilder->new_from_url($site);
   my @quotes;

   for ( $tree->look_down( _tag => 'p' ) ) {
      if ( ( my $t = $_->as_text ) =~ m{ ^ \d+ \. \s+ }x ) {
         $t =~ s{ \x{2019}  }{'}gx;
         $t =~ s{ \xA0      }{ }gx;
         $t =~ s{ \x{2013} }{--}gx;
         push @quotes, $t;
      }
   }

   my $randomelement = $quotes[ rand @quotes ];
   print "$randomelement\n";
   $randomelement =~ s/^\d+\.\s+//;
   return $randomelement;
}
[download]

How would a person represent an appointment in such a scheme?

The co-ordinates start bottom left of page. Start a new page when your line position gets to the end of the page like this.

for my $item (@task) {
  my $msg = '[  ]  ' . $item;
  drawline( $page, $linepos );
  $txt->translate( 70, $linepos+10 );
  $txt->text($msg);
  $linepos -= $vspace;
  if ($linepos < 0){
    $page = $pdf->page;
    $txt  = $page->text;
    $txt->font( $font, 20 );
    $linepos = 650;
  }
}
[download]

You might consider a newpage sub to repeat the heading and add a page number.

How would a person represent an appointment in such a scheme?

Sorry but I don't understand the question

poj

How would a person represent an appointment in such a scheme?

in a first draft, you could add it to @task but you'd have to guess for the right position.
Most calendars have different areas (e.g. columns) for tasks (without particular time) and events (on a time scale). I'd go for that, too.