You could do something like this:

my $logger;
use constant DEBUGGING => !!$ENV{DEBUG};
sub log_this (&) {
   DEBUGGING or return;
   $logger ||= Logger->new;
   $logger->log($_) for shift->();
}

log_this { "Hello world" };
log_this { "Hello", "world" };
log_this {
   my $data = Expensive->method_call();
   "DATA: $data";
};
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If the $DEBUG environment variable is false, log_this will return quickly without evaluating the contents of the code block. This still has the overhead of the sub call to log_this itself, but that overhead will be fairly small.

As a bonus, you could use something like Keyword::Simple to alter the parsing of log_this to avoid having even the overhead of that redundant sub call when $DEBUG is false. If you write your custom parser carefully, so that it has no real effect on the syntax of the log_this keyword but just causes it to be no-opped as appropriate, then the custom parsing could be dropped in for a speed-up only on those versions of Perl that support it (5.14+), but it would still work on older Perls.