Like Moron said, parsing XML with regexes is a pita and error prone. The code below uses XML::XPath to do the same job yours does. And besides from being safer it's also more readable (and more perlish :-).

Note: I added a root element to the data so it becomes valid XML.

use strict;
use warnings;

use XML::XPath;


my $xp = XML::XPath->new(ioref => *DATA);

#for parsing files:
#my $xp = XML::XPath->new(filename => 'test.xml');    
    
print map  { $_->string_value, "\n" } 
      grep { $_->string_value       }
      $xp->find('/Root/Table')->get_nodelist;
__DATA__
<Root>
    <Table>First_Table</Table>
    <Table/>
    <Table>Second_Table</Table>
</Root>
[download]

Ouputs:

First_Table
Second_Table
[download]