Use the first form if you are mostly interested in how those things affect your car or your ability to drive. Use the second form if you are more interested in the tires or gas tanks themselves. My car's gas tank was empty, so I had to stop driving and buy gas. My car's tires were larger than Bob's car's tires were, so my car drove better on gravel roads. At this store, I sell bike tires and car tires. My bike tires are near the back of the store, and my car tires are near the front. In some sentences, you are approximately equally interested in your car and its tires. In these sentences, both patterns are OK: Both OK: I need to inflate my bike('s) tires more often than I need to inflate my car('s) tires.

It is more correct to write it the first way in both cases: "My car's gas tank" and "My car's tires". However, if you write it the second way, it would be understood.

Chris' answer is excellent.