I think your problem is the interpolation and non-interpolation with double versus single quotes. You don't need to escape your backslashes and dollar sign in your %typehash as you're using single quotes.

This worked (printed the right path) for me:

#!C:\perl\bin\perl

use strict;
use warnings;

my %typehash = (
    'tomcat'    => '\d$\logs\tomcat',
    'tomcatweb' => '\d$\logs\web',
    'apache'    => '\d$\logs\Apache'
);

sub getLog { 
    my ($a,$type)=@_;
    my $server=$a;
    my $dir =  "\\\\$server".$typehash{$type};
    print "\n $dir ";
    chomp(my $meh = `dir /b /O:D $dir`);
    print $meh;
} 

&getLog("test.control.com",'tomcat');
[download]

And executed:

{Z} \\vmware-host\Shared Folders > perl script.pl

 \\test.control.com\d$\logs\tomcat The network path was not found.
[download]

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