What is the name of the class of French-language rhymed couplets in
which the entire first line sounds just like the entire second line,
except that the lines have two different meanings and the two lines
together make up a meaningful couplet?  Also, where can I find
examples of these on the Internet?

Dear nyken,

This kind of short poem is called "poème holorime", "vers holorimes",
or simply "holorime". Other names are "vers olorimes", "vers
millionnaires", and "vers homophones".

Here are some websites with examples of French poèmes holorimes:

Fatrazie: Holorimes
http://worldserver2.oleane.com/fatrazie/Holorimes.htm

Philolittérement: Vers olorimes
http://pages.infinit.net/plotin11/olorimes.htm

Le Bric-à-brac de Sylvain: Vers (h)olorimes 
http://www.bric-a-brac.org/lettres/olorimes.php


Sources:

Wikipedia Online Encyclopedia: Poème holorime (in French)
http://fr.wikipedia.org/wiki/Po%E8me_holorime

Everything2: Holorime
http://www.everything2.net/index.pl?node_id=1324724&lastnode_id=655029

Search terms used:
holorime
://www.google.de/search?q=holorime&ie=UTF-8&oe=UTF-8&hl=de&meta=
holorimes
://www.google.de/search?q=holorimes&hl=de&lr=&ie=UTF-8&oe=UTF-8&newwindow=1&start=0&sa=N
vers olorimes
://www.google.de/search?hl=de&ie=UTF-8&oe=UTF-8&newwindow=1&q=vers+olorimes&btnG=Google+Suche&meta=

Special thanks to my dear colleague pinkfreud-ga for reviving my memory.

Hope this answers your question!
Regards,
Scriptor

just what I needed!