You can pass variables and file handles to subroutines. See perlsub for details. Here is an example:

use strict;
use warnings;

my %function = (
    allergies       => \&print_allergies,
    immunizations   => sub {
        my ($fh, $dir) = @_;
        print "\nin immunizations from directory $dir\n";
    },
);

my $dir = 'input_files';
opendir DIR, $dir or die "$dir: $!";

while (defined(my $file = readdir DIR)) {
    next if($file =~ /^\.+$/);
    print "The directory and file are $dir/$file\n";
    open my $fh, '<', "$dir/$file" or die "$dir/$file: $!";
    $function{$file}->($fh, $dir) if exists $function{$file};
}
closedir DIR;



sub print_allergies{
    my ($fh, $dir)  = @_;

    print "\nprinting allergies from directory $dir\n";
    while (<$fh>){
        next if (/^#/);
        print  $_ ;
    }
}
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update: added example of passing a variable other than a file handle.