When shoes wear out, the biomechanics of running do change. Just not much. Kong et al tested 24 runners before and after 200 miles of road-running in the same pair of shoes. There were a few minor changes: longer stance phase, less forward leaning, and less ankle flexion. However, hip and knee angles were unchanged.
[Update: several runner-readers have commented that 200 miles is not much, and they would hope that shoe wear would not have significant effects on the biomechanics of running until at least 300 or 400 miles. It’s possible that the relatively small number of miles is a bit of a deal-breaker for this study. ~ Paul 2016-02-16]
It’s significant that the effects on gait were generally minor, of course, but note the lack of difference in knee angles particularly. Knees are the site of two of the most common runner’s knee injuries, both iliotibial band and patellofemoral syndrome. In general, forces and injury risks increase together with more movement — more bending puts more torque on joints. A deeper knee bend is more stressful than a shallower knee bend. If worn out shoes have no effect on how far knees bend as you run, they aren’t much cause for concern. Although there are surely other biomechanical factors involved in these injuries, it’s unlikely that they are affected much by the condition of your shoes, and it’s reassuring that the most significant factor in overall knee stress — degree of flexion — is probably entirely unaffected.
Also interesting: they also compared kinds of shoes — air, gel, spring — and concluded “the adaptation strategies to shoe degradation were unaffected by different cushioning technologies.” So it makes no difference what kind of common cushioning method you have in your shoes — they all degrade and affect running about equally. A natural enemy of the salesman is science proving that there is no difference between products!
Nevertheless, the precautionary principle applies, and I do still recommend replacing your shoes when they begin to show obvious signs of wear. The risk of running in decrepit shoes may be small, but there’s not much reason to take that risk — just the modest cost of buying shoes somewhat more often. It’s not like you weren’t going to buy new shoes eventually! On the other hand, this data makes it pretty clear that replacing shoes while they still look fine isn’t really going to make much of a difference.