Since I've spent a few hours tracking down the issue, I'm posting the results in case someone else finds this useful.
The issue manifests itself as
(1) Library Browser does not update problem counts when 'subject' is selected, with 'chapter' and 'section' lists not being populated
(2) The web browser console reports
"SyntaxError: JSON.parse: unexpected character at line 1 column 1 of the JSON data"
(3) The web server log files have the following error messages:
Can't use string ("") as a subroutine ref while "strict refs" in use at /usr/local/share/perl/5.14.2/SOAP/Transport/HTTP.pm line 384
[/webwork2/instructorXMLHandler] Can't call method "result" on an undefined value at /opt/webwork/webwork2/lib/WebworkClient.pm line 164.\n * in WebworkClient::xmlrpcCall called at line 318 of /opt/webwork/webwork2/lib/WeBWorK/ContentGenerator/instructorXMLHandler.pm\n * in WeBWorK::ContentGenerator::instructorXMLHandler::pre_header_initialize called at line 183 of /opt/webwork/webwork2/lib/WeBWorK/ContentGenerator.pm\n * in WeBWorK::ContentGenerator::go called at line 370 of /opt/webwork/webwork2/lib/WeBWorK.pm
The server log messages turn out to be the most useful ones. There is something wrong in the SOAP::Transport::HTTP module; most of it appears to be a leftover code from development/debugging.
Downgrading to SOAP::Lite version 1.11 resolves the problem; one can do that in CPAN shell with the following command