Do you want it to be silent for the redirection? Use a simple_request instead of request.

my $ua = LWP::UserAgent->new;
my $uri = shift @ARGV;
my $headreq = HTTP::Request->new(HEAD => $uri);        
my $headres = $ua->simple_request($headreq);   # <----
my $statuscode = $headres->code();        
print "Status 401 at $uri\n" if $statuscode == 401;
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Or do you wish for it to display the uri to which you were redirected? Access it through the Request that produced the Response.

my $ua = LWP::UserAgent->new;
my $uri = shift @ARGV;
my $headreq = HTTP::Request->new(HEAD => $uri);        
my $headres = $ua->request($headreq);        
my $statuscode = $headres->code();        
if ($statuscode == 401) {
   my $final_uri = $headres->request()->uri();   # <----
   print "Status 401 at $final_uri\n" 
}
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