How can I calculate if it makes sense from a resource-conservation perspective to purchase a new, more energy efficient appliance, and when it makes sense to use an old appliance until it is beyond repair? Are there appliances that are commonly known to be one or the other? I am interested in the environmental impact, not in the economics.
Let's break down some of the elements of whole-life costing, to glean some clues about when old might be better, and when new might be better.
If the manufacturing impact of a new device is a high proportion of total lifecycle impact, and if there is a chance that the impact of the latest model will lessen within the extended lifetime of the old device, then repairing the old device is likely to be the better option, giving industry time to clean up its act before you buy a new model.
If the environmental impact of running the old device represents a large proportion of the whole-lifecycle impact, as it does for cars, then that would typically favour buying a new model.
This is where the cradle-to-cradle philosophy becomes essential: the whole-lifecycle cost includes the cost of turning an old dead device, into components for new devices.
If you have a good way of disposing of the old device, such that most of its material will have a very low-impact route to reuse, then buying a new device may be the better option. So, for example, if you're going to trade in an old vehicle for a new one, and the garage you trade in at is going to strip down the old vehicle and use all the mechanical bits for spares, and all of the other materials will be recycled, then that would favour buying a new vehicles.
On the other hand, if disposing of an old dishwasher is going to lead to it being dumped in landfill, then the cost of extracting it back out of landfill and reusing the bits, and decontaminating the land afterwards, could be quite high, and that would favour repairing the old model.
A closing note on the economics
You have quite reasonably said in your question that you are interested in the environmental impact, and not in the economics. I thought it might be useful to just note note for other readers why there is a difference between those two, and why they matter differently.
In a perfect world, the economics would tell you everything you needed to know about the sustainability. If all externalities were priced in, and if the market were efficient, then the price would reflect everything you needed to know to answer this question - it would include whole-lifecycle costs, on a cradle-to-cradle basis, for each, using a sane discount rate.
Sadly, there remain many unpriced externalities; markets are often very inefficient; whole-lifecycle cradle-to-cradle costs can be difficult to discover; and people disagree on what a sane discount rate; so the economics can be a very poor guide, unfortunately.
There's also a problem that there isn't a single universal standard for "environmental impact": there are sometimes trade-offs to be made between greenhouse-gas emissions, freshwater usage, local pollution, and so on - it's very difficult to even agree the damage-cost function for a unit of freshwater pollution, because that will vary by time and place.
All of these gaps between the perfect market and markets as they are, create inefficiencies, and cultivate unsustainability and embed it into our economy. So, sometimes it can be more economically efficient to let what we know about environmental impact override what we observe as the market price: because there are areas where we know more about costs than what the market has priced in.
The energy embodied in durable goods pretty much guarantee the greatest environmental benefit is derived from keeping the old until it dies. That includes performing repairs up to the actual cost of a new unit. Many, especially older, durable goods are modifiable at the time of repair to make them more eco-friendly. This is always the most sustainable practice. If it'd just go to the landfill or recycling plant it's almost always a sustainability loss to replace.
A few durable goods actually have pretty low embodied energy. In these cases the most sustainable practice actually is replacement.