And the title asks, if a character with mirror images cast invisibility (or vanish) do the images become invisible too?
Yes. Here's why. If a character with images up is affected by fireball and takes damage, all his images also appear to take damage (including being visibly singed. If a character with images up pulls out a rod of wonder and uses it (and as a result turns himself blue), all of the images appear to pull out a rod of wonder, use it, and turn blue.
You managed to cast mirror image on your thread, which is pretty awesome. ;) Anyway:
The spell description itself answers your question, albeit somewhat indirectly:
An attacker must be able to see the figments to be fooled. If you are invisible or the attacker is blind, the spell has no effect (although the normal miss chances still apply).
This means if you are invisible, your figments cannot be seen--ergo, they also become invisible.
As a GM I would rule that the mirror image spell is not dispersed, however, but that the duration would run out as normal--simply that while you are invisible, the images cannot be attacked in your stead.