It seems that ->readFromFileHandle really wants the filename. The following works for me:

use strict;
use LWP::UserAgent;
use Archive::Zip;

use Archive::Zip qw( :ERROR_CODES :CONSTANTS );
use Data::Dumper;

my $url = 'http://keep.drewhead.org/test.zip';
my  $file = '../test.zip'; # a local copy of $url

 
# This doesn't
{
  my $ua = LWP::UserAgent->new;
  $ua->timeout(10);
  $ua->env_proxy;
  my $response = $ua->get($url);
  warn length($response->decoded_content);
  
  my $tmp;
  if ($response->is_success) {
      $tmp = File::Temp->new();
      binmode $tmp;
      print {$tmp} $response->decoded_content;
      print "Wrote ".$tmp->filename()."\n";
  }
  else { die "error ".$response->status_line; }
  
  #unless (-B $tmp) { die "Did not get a filehandle\n"; }

  my $zip = Archive::Zip->new();
  my $zip_err = $zip->readFromFileHandle( $tmp, $tmp->filename );
  unless ($zip_err == AZ_OK ) {
    die "Archive::Zip read error on filehandle: $zip_err\n";
  }
  
  foreach my $member ($zip->members()) {
    my $fileName = $member->fileName();
    print "fileName = $fileName\n";
    my $content = $member->contents();
    print "contents $fileName = ".Dumper($content);
  }
}
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I added the binmode call before writing the tempfile, and I pass the name of the tempfile down to ->readFromFileHandle().