500 Usage: $h->push_header($field, $val)

After upgrading the LWP::UserAgent module to version 5.824, I started getting a bunch of errors like 500 Usage: $h->push_header($field, $val) in the error logs. SOAP::Lite seemed to be the top-level cause of the error. It turns out that I had an old version of SOAP::Lite (version 0.69). Upgrading to version 0.710.8 fixed the problems.