Organisations such as JAK and Rentevrij argue that economic interest is fundamentally unsustainable, because it requires perpetual exponential economic growth, something impossible in a finite ecological system.
Can interest and sustainability co-exist in the long run?
Yes, interest and sustainability can co-exist in the long run. Interest has already survived for millennia, so it's probably one of the best-tested sustainable mechanisms we have as a civilisation.
And interest does not require perpetual exponential growth, neither for savings nor for debt. We've had plenty of periods of history where we haven't had exponential economic growth, and we have had interest.
Don't confuse money with resources: they are exchangeable, at always-varying rates, but they are not the same thing. Money can be created at the push of a Central Banker's button - that's what Quantitative Easing is. And debt / savings can be eliminated at the push of a button too - that's what debt cancellation is. And both of these things are happening around us right now.
Furthermore, interest represents a real price - the price of credit. If you've got a sum of money that you can access now, you can choose to spend it now, or you can choose to spend it later. If you've got the same sum of money, but you can only access it in one year's time, then you can spend it any time after 1 year, but at no point between now and then. So, by definition, that money that is locked away for a year is worth less than money available now, because you can do less with it. And the price of that is what interest represents - the price of credit.
Credit has a real price, and to hide that real price, would distort the market for credit, creating long-term distortions, and actually worsening economic efficiency and thus economic sustainability. Now, economic sustainability is by no means the only kind - there's social sustainability, resource sustainability, and so on. And they are all important.
If I understand you correctly, you are mixing two concepts. Sustainable living can exist regardless of the economic climate. Prices rising does not mean that the value of assets is rising. It mean that you need more paper money to buy those assets.
I am no economist, but at the moment it seems that the central banks are charging interest on the money they give out. Which means they will have to print more money to give to you to pay the interest. But this money again will have interest on it... This setup leads to more money is accounted for than actually exists until the central bank decides to cancel your debt, for services, assets, or just because they like you.
Now, the finiteness assumption about this earth's resources could probably lead to a more sustainable model of economy, but this is a different question, and probably more suited for Economy SE (which seems not to exist any longer).
In response to @EnergyNumbers's comment, I should add that interest your normal bank charges can exist in a sustainable model of economy. Charging for the service of lending you money is just like charging for cooking a meal. In that sense interest is perfectly sustainable (in the economic sense).
