Nope. That doesn't work either. I wrapped your code into a sub, so I could call it multiple times and got:

Testing(some data) at D:\test.pl line 5, <DATA> line 1.
Testing at D:\test.pl line 13.
Testing(some data) at D:\test.pl line 5, <DATA> line 1.
Can't save DATA: Bad file descriptor
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Here is test.pl:

sub Tye
{    
  my $pos= tell(DATA);
  chomp( my $x= <DATA> );
  warn "Testing($x)";
  {
    my $fh= do { local(*FH); \*FH };
    open $fh, "<&DATA"  or  die "Can't save DATA: $!\n";
    close DATA  or  warn "Can't close DATA: $!\n";
    open DATA, "<&=".fileno($fh)  or  die "Can't restore DATA: $!\n";
    seek DATA, $pos, 0  or  die "Can't reposition DATA: $!\n";
  }
  warn "Testing";
}

Tye();
Tye();
__END__
some data
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