<p>Is it okay to run Hibernate applications configured with <code>hbm2ddl.auto=update</code> to update the database schema in a production environment?</p>

No, it's unsafe.
Despite the best efforts of the Hibernate team, you simply cannot rely on automatic updates
in production
. Write your own patches, review them with DBA, test them, then apply them manually.
Theoretically, if
hbm2ddl update
worked in development, it should work in production too. But in reality, it's not always the case.
Even if it worked OK, it may be sub-optimal. DBAs are paid that much for a reason.