It seems that despite what the docs say, you have to use the descriptive name to set the font.

Anyhow a better way than destroying and recreating the HList widget, is to clear the contents and than add new contents like in the following example:

use strict;
use warnings;

use Tk;
use Tk::HList;
use Tk::ItemStyle;
use Tk::FontDialog;

my $mw = MainWindow->new;

$mw->title("View");
$mw->minsize(560, 180);

my $font = '{Architects Daughter} -12 bold'; # have to use descriptive
+ name

my $top
    = $mw->Frame()->pack( -side => 'top', -fill => 'both', -expand => 
+0 );
my $hlistframe
    = $mw->Frame()->pack( -side => 'top', -fill => 'both', -expand => 
+1 );
my $bottomframe = $mw->Frame()->pack( -fill => 'x', -expand => 0 );

my $menubutton
    = $top->Menubutton( -text => "Font", )->pack( -side => 'left' );
$menubutton->command(
    -label   => "Change",
    -command => sub {
        $font = $mw->FontDialog->Show;
        $mw->RefontTree( -font => $font ) if defined $font;
        print "Font changed: $font\n";
    }
);

my $hl = $hlistframe->Scrolled(
    'HList',
    -scrollbars => 'ose',
    -columns    => 2,
    -header     => 1,
    -height     => -1,
)->pack( -fill => 'both', -expand => 1 );

my $label1 = $hl->Label( -text => "Tool", -anchor => 'w' );
$hl->headerCreate( 0, -itemtype => 'window', -widget => $label1 );
my $label2 = $hl->Label( -text => "Vendor", -anchor => 'w' );
$hl->headerCreate( 1, -itemtype => 'window', -widget => $label2 );
#...

$bottomframe->Button( -text => "Add item", -command => \&add_item )
    ->pack( -side => 'right', -anchor => 'ne', -fill => 'x' );

$bottomframe->Button( -text => "Clear", -command => \&clear )
    ->pack( -side => 'right', -anchor => 'ne', -fill => 'x' );

$mw->RefontTree( -font => $font ) if defined $font; # set default font

my $style = $hl->ItemStyle(
    'text',
    -background => 'light green',
    -font       => $font,
);

add_item();

sub add_item {
    my $e = $hl->addchild('');
    if (defined $font) {
        my $font_descriptive = $mw->GetDescriptiveFontName($font);
        $style->configure( -font => $font );
        print $font_descriptive, "\n";
    }
    $hl->itemCreate( $e, 0, -text => 'Col0', -style => $style );
    $hl->itemCreate( $e, 1, -text => 'Col1' );
}

sub clear {
    $hl->delete('all');
}

MainLoop;
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This way changing the font of the widget is not needed anymore.