I spent 17 years in Alaska. I've driven a combined thousands of miles on dirt and gravel roads. I think that's a slight exaggeration.

A true dirt track of a road like the McCarthy Road that contains leftover railroad spikes and things can easily puncture a tire, but that's a hundred or so bucks. I'm not sure how you'd get $5,000 in damage on that road unless you drove over an exposed rock at exactly the right place to pop a hole in your oil pan and then kept the engine running after the oil had all drained out.

A mere gravel road, though? Zero chance of damage. Maybe some long-term effects on the car's suspension if you drive thousands of miles on one that's not well graded, but stay well back from other cars and especially trucks (to avoid a pebble cracking your windshield) and you'll be fine.

I worked in the rental car industry in Alaska. We weren't at all concerned about someone driving five miles on a dirt road to access a trailhead. We just didn't want them taking off 500 miles down one to Prudhoe Bay or across the Denali Highway or on the aforementioned McCarthy Road.

If you do go on a dirt road, just do two things:

1) If your car breaks down and you have to call for help, nurse it back to a paved highway before you have the rental car company dispatch a tow truck.
2) Consider washing your car before you return it to remove evidence that you went off of paved road.

If you keep those two things in mind, you can drive with impunity on as many dirt roads as you want. The rental company will never know.