Well, that's not too bad. The only obvious things that might make it more perlish (imho) would be to use the implicit argument ($_) rather than $line, and to process the file in chunks other than lines. I also wouldn't use FileHandle; it's not necessary in modern Perls.

Here's a first clean-up, which uses a hash index rather than a reference to a hash value:

local $ARGV = ($templfile);
my $i = '';
while (<>)
{
    if ( /^\s*<\s*TEMPLATE\s+NAME="(.*)"\s*>/i )
    {
        $i = $1;
    }
    elsif ( /^\s*<\s*\/\s*TEMPLATE\s*>/i )
    {
        $i = '';
    }
    else
    {
        $hashref->{$i} .= $_;
    }
}
# cleanup:
delete $hashref->{''};
[download]

Here's another approach which is pushing the idiomaticity:

while (<>)
{
    no warnings 'uninitialized';
    /^\s*<\s*TEMPLATE\s+NAME="(.*?)"\s*>/i
    ..
    /^\s*<\s*\/\s*TEMPLATE\s*>/i 
    and $h{$1} .= $_;
}

# cleanup
delete $hashref->{''};
s/^\s*<\s*TEMPLATE\s+NAME="(.*?)"\s*>\s*//i 
    for values %$hashref;
[download]

Beyond that, you could do things like

local $/;
local $_ = <>;

%$hashref = /\s*<\s*TEMPLATE\s+NAME="(.*?)"\s*>(.*?)\s*<\s*\/\s*TEMPLA
+TE\s*>/sig;
[download]

but then you have to worry about the size of the file in memory.