what are the implications of setting up a subscriber as a publisher? We have application database (APP_DB) that creates a "live copy" of itself via transactional replication (push) to a database (COPY_DB) on a separate server.  This COPY_DB is used for reporting and troubleshooting.

There is now a need for a small subset of the tables (4 tables out of 150) to be placed ("live") in a second database (NEW_DB).  What would be the implications of setting up COPY_DB as a publisher of these four tables to subscriber NEW_DB?  Good idea?  Horrible idea?

It seems like a silly idea to me, but I find myself in a situation where I have more control over the server where COPY_DB and NEW_DB live and could more easily implement such a solution rather than setting up a second publication on APP_DB.
This is known as the republishing model.  It is usually used for balancing distribution loads over slow or expensive links although I have seen it used in the scenario you described as well.  The republishing topology you propose is supported which can be found near the bottom of the page Republish Data, Transactional publication -> Transactional subscription/transactional publication -> Transactional subscription.

If I recall correctly, the challenge with this topology occurs if you ever have to reinitialize COPY_DB, you will also have to reinitialize NEW_DB.