Most Australian car rental companies will say you are not supposed to take the rented car on unsealed surfaces.

Check if the road is unsealed and check with the company on their specific conditions and if this general restriciton also applies to 4 wheel-drives. I think 4WDs might be exempt.

http://www.carhireinsider.com.au/where-you-can-drive-hire-cars/

Found this link which I think you might find quite useful.

A section of the road remains unsealed. Ask the company directly and get it in writing if they say yes. If they say no, go to another company.

Thanks, everyone. I've emailed the rental company and hope to get a specific answer. Speaking of the road to Chillagoe, is it reasonable to think that we could go there and back in one day from Cairns? My son has just informed me that we cannot go to Australia without seeing the Outback, and I think that's as close as we're going to get. Unless you guys have a better idea!

You could do it in one day if you are happy to spend the entire day driving (7 hours driving time). I would overnight in Chillagoe for at least one night so you can see some of the caves.

Why not hire a 4WD and take the back road from there to Undara. The road is not too taxing for those without experience and the new experience will be a lasting memory. Then you will have really been out back. Undara has a wealth of wildlife as well as the largest and longest lava tubes on earth.

What I am suggesting above is a three day trip but really you could then stop on the Tablelands to see the sights and the wildlife there and make it a four day trip.

The unsealed portion of the road to Chillagoe is pretty good – I rode it some years ago on a bicycle, at night. Hard pedalling, but otherwise OK. I think that there was about 10km of unsealed road, and the marble quarry was quite spooky in the dark, white monoliths.

Undarra is really worth a visit. I was expecting the lava tubes to be like a rabbit burrow – and was amazed to find caves big enough to drive a small truck down. The caves are a micro-environment, and you can walk into some of the caves for about 500 metres. A whole ecology – bats, albino insects that eat the bat droppings, and unique parasites that live on the insects.

There is a range of accommodation at Undarra.

I would endorse Undarra Lava Tubes too, as well as the Atherton Tablelands. We did a one day each trip with a tour company some years ago, and it was amazing. If you drive yourself you'll be able to concentrate on what you are more interested in.
Enjoy your trip. It's another interesting corner of Australia I enjoyed getting to know albeit briefly.