No need for recursion. Just change the anonymous sub as follows (untested):

my $fh_iterator = sub
{
    my $fh    = shift;
    my $line  = $fh->getline();
       $line .= $fh->getline() while $line =~ m{\\$};

    return $line;
}
[download]

Update: Here is a tested script which eliminates the Use of uninitialized value warning reported in the post below:

use strict;
use warnings;
use IO::File;

my  $filename = shift @ARGV;
my  $fh       = IO::File->new($filename, 'r');

sub fh_iterator
{
    my $fh   = shift;
    my $line = $fh->getline();

    if (defined $line)
    {
        $line .= $fh->getline() while $line =~ m{\\$};
    }

    return $line;
}

while (my $line = fh_iterator($fh))
{
    print $line;
}
[download]

Output:

16:29 >perl 738_SoPW.pl test.file
foo \
bar \
baz
single line

16:29 >
[download]

Hope that helps,