As per Special Literals:

The program should close DATA when it is done reading from it. (Leaving it open leaks filehandles if the module is reloaded for any reason, so it's a safer practice to close it.)

So yes, any module reading from its DATA should close it. BTW, it's not Geography::Countries::LatLong at fault here, it doesn't have any __DATA__ sections, it's its parent, Geography::Countries. A bug/patch could be filed with that module, although it appears unmaintained.

As for the general feature of appending the information to the error/warning messages, personally, I think it's a useful feature. I believe what it's using is the internal "last filehandle read", the same that eof and tell use when called without arguments, and that $. uses; see e.g. Filehandle, last accessed. Although not extra pretty, and maybe I'm missing an easier way to do this, this works:

use warnings;
use strict;

sub resetfh { eof do { local *HANDLE; *HANDLE } }

my @x = <DATA>;
resetfh;
warn "test";

__DATA__
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