A bit more context would help people help you.

For example, what does substemplate() do?

As a guess, I'd say you should to reorganize your code into a loop, and test the strings before calling substemplate(); here's an untested example of what that might look like:

    my $templ=opentemplate('regform','normal');
    substemplate($templ,'fullname',$arr[0]);
    substemplate($templ,'header',$arr[3]);
    #
    $link0="<a href=http://$arr[2]>$arr[1]</a>";
    substemplate($templ,'link0',$link0);
    #
    foreach my $linknum ( 1 .. 10 ) {
      my $pos = $linknum * 2 + 2;
      my ( $text, $url ) = @arr[ $pos, $pos + 1 ];
      my $link = ( ! $text or ! $url ) ? '' :
        "<tr><td><a href=http://$url>$text</a><br><br></tr></td>";
      substemplate($templ,"link$linknum",$link);
    }

    substemplate($templ,'refer',$refer?$refer:'none');
    print $templ;
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