Didn't work using an always fail function with any of the following tests:

sub check_name {
    return 0;
}


unless (&check_name($app{"first_name"})) { # code } # ,
unless (&check_name($app{"first_name"}) == 1) { # code } # ,
unless (&check_name($app{"first_name"}) eq 1) { # code } # , or
unless (&check_name($app{"first_name"}) eq "1") { # code }

# perl -v
# This is perl, v5.8.4 built for MSWin32-x86-multi-thread
# (with 3 registered patches, see perl -V for more detail)
[download]

Try running this on your commandline:

use warnings;
use strict;
use Data::Dumper;

my @errors;
my %app;
$app{"first_name"} = param('first_name');
# $app{"first_name"} = "test_value";

if (request_method() eq "POST") {
    my $first_name_check = &check_name($app{"first_name"});
    unless ($first_name_check eq 1) {
        print "Not through namecheck\n";
        $app{"first_name"} = "";
        push (@errors, "First Name: ".$first_name_check);
    }
}

print Dumper \@errors;
print Dumper \%app;

sub request_method { return 'POST' }
sub param { return 'testname' }

sub print_application {
    # do some stuff with %app and @errors...
}

sub check_name {
    my $name = $_[0];
    if ($name =~ /^[A-Z][A-Za-z\-']{2,40}/s) {
         return 1;
    } else {
         return "Helpful message.";
    }
[download]

I got

Not through namecheck
$VAR1 = [
          'First Name: Helpful message.'
        ];
$VAR1 = {
          'first_name' => ''
        };
[download]

-- 
Thru the darkness of futures past, the magician longs to see.
One chants out, between two worlds - fire, walk with me.

Works for me on the command line. Also works using most recent changes copied out of the script and into your test:

unless (&check_name($app{"first_name"})) {
    undef $app{"first_name"};
    push (@errors, "Helpful message...");
}
[download]

It still doesn't work in CGI. I added use Data::Dumper and to the actual script and "Dumper" to the end of the &print_application function:

sub print_application {
    header,
    start_html,
    table({-border=>'0' -class=>'boxed'},
    Tr(
        [
            td([b('First Name:'), "$app{'first_name'}"])
        ]
    ),
    p("Value is", $app{'first_name'}),
    pre(
        Dumper \@errors,
        Dumper \%app
    ),
    end_html;
}
[download]

Oddly enough, this is what comes out the other side:

A form element with the value "nameCheckFailsIfNumbersPresent12345" in
+ a table, and...

Value is  # <- undef here has it should be. But not above?

$VAR1 = [
          'Names must be between two and forty total characters, start
+ with an initial capitol, and may only contact alphabetical character
+s, a dash, or an apostrophe.'
        ];
 $VAR2 = '$VAR1 = {
          \'first_name\' => undef
        };
';
[download]

That looks like it works, except this line from the same print_application function: print textfield(-name=>'first_name', -size=>'40', -maxlength=>'40', -default=>"$app{'first_name'}");

returns... <input type="text" name="first_name" value="nameCheckFailsIfNumbersPresent12345" size="40" maxlength="40">

Why is the value "nameCheckFailsIfNumbersPresent12345" present in $app{'first_name'} but not in Dumper \%app or p("Value is", $app{'first_name'}) - all of which are called from inside the function?

Did this for fun and the value shows in the textfield but not as the plain variable. Also tried removing the double quotes surrounding $app{'first_name'} in textfield().

Tr(
    [
        td(
            [
                b('First Name:'),
                textfield(-name=>'first_name', -size=>'40', -maxlength
+=>'40', -default=>"$app{'first_name'}")
            ]
        ),
    ]
),
Tr(
    [
        td(
            [
                b('First Name:'),
                $app{'first_name'}
            ]
        ),
    ]
);
[download]

Those users sure better appreciate not having to re-type form data... :-)