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.

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... :-)

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?