I don't think there's a point to running a car into the ground just for the sake of doing so, nor do I think it's the optimal strategy to reduce costs.
Buying a new car is not the answer (though current tax incentives might make me consider a new electric car if I was driving a lot), but I think the sweet spot lies in buying a slightly used car, using it for a moderate amount of time, and then selling it before you have too many maintenence costs and headaches but while it still has some resale value.
Ideally you'd know the depreciation and maintenance curves over time for the specific car that you are dealing with; but since you can't know that before the fact, averages can give you a general idea. There is nice data from
consumer reports:

You can kind of speculate from their data that around year 6 or 7, the decrease in carrying costs per year is offset by the increase in operating costs. That is all based on averages across all vehicles and driving styles. If you have a car that costs less to own to start off with, and you drive less, then it would shift the sweet spot to the right. If you got stuck with a car that had a lot of mechanical problems, the sweet spot would be further to the left (but you'd only know that after the fact).
It's kind of interesting that they single out the Honda Fit in that report as a particularly cheap car to own overall: "It combines a relatively low purchase price with low depreciation, great fuel economy, excellent reliability, and fairly low maintenance and repair costs."
Edmunds has their own graph, which seems based on a lot less robust data, but they also put their sweet spot at about 5-7 years, considering depreciation only:

The downside to all this is transaction costs, and the difficulty (pain in the ass) of finding a good quality reliable used car.. but if you are having trouble with your current car and not happy with it, you might have more motivation to consider car shopping.
For reference, we have a 2009 Honda Fit, which we bought used at about the two year old mark. If we were driving a lot, or starting to encounter maintenance issues, I'd probably consider selling it about now and buying something like a 2013 - 2015 Fit. We don't drive much, and are happy enough with our car that I don't feel like dealing with the hassle of transactions costs.
So the real answer is to not drive much, or at all, but there might be a second best answer that involves selling the car before the point that it is a wreck.