The question has the tag "mountaineering," but most of the time when I hear people say that you need boots with ankle support, they're actually talking about trail-walking. The cases of hiking and mountaineering are qualitatively different.
For mountaineering, one big reason people usually don't use lightweight running shoes is that often there is talus, scree, and loose rock. Small scree is a nuisance if it gets inside your shoes, although there are lightweight gaiters with velcro attachments (e.g., Dirty Girl) that do a great job of keeping that out. The bigger issue is that on a steep scree or talus slope (the kind of thing you have to scramble up, not a trail), there is often unstable rock, and it can be nice to have your feet protected against getting banged or crushed. When coming down a scree slope, boots allow you to do a scree glassade, which is much faster and infinitely more fun than the alternative.
For trail walking, the idea of wearing heavy boots for ankle support is at best a matter of personal preference, and at worst a harmful superstition. Plenty of people carry heavy packs over uneven ground while wearing running shoes. An influential advocate of lightweight shoes for this purpose was Ray Jardine, who wrote the 1992 book Beyond Backpacking. IIRC Jardine describes in the book how he goes on long-distance backpacking trips with his girlfriend (doing something like 25 miles a day), and he uses running shoes while she uses boots; he portrays it as a matter of personal preference. The problem with boots is that they're heavy, and weight on your feet hurts your efficiency many times more than an equivalent weight carried on your back. Also, the heavier your footwear, the harder it is to get a comfortable fit and the more likely you are to get blisters.
Running shoes do become impractical in very specific situations. They aren't good in snow, for slogging through deep mud, for technical rock climbing, or for very cold conditions. Sometimes if you have a day of hiking with 20 or 30 little stream crossings, it can be a real nuisance to use running shoes; if you're wearing water-resistant boots, you can walk straight through a shallow stream.
Running shoes don't work well if there's a lot of snow (although it can be done -- a lot of PCTers get through the Sierra in early season using running shoes). And running shoes are obviously not compatible with crampons, although it is possible to throw a pair of microspikes in your pack if you think you might just encounter a few isolated icy patches on a steep trail.
For people who are afraid they'll turn an ankle if they try to use running shoes with a heavy pack on uneven ground, there are still alternatives to boots. One is simply to keep doing the activity in running shoes, which will strengthen your ankles until you feel more physically confident. Another is to adopt a lightweight or ultralight style of backpacking, so that you're not tottering down the trail with the Leaning Tower of Pisa on your back. Jardine's book was an early influence in establishing the concept of ultralight backpacking, but today a better source of information is probably the web site backpackinglight.
Four mountaineering, many people carry a separate pair of lightweight approach shoes that are more comfortable than mountaineering boots. They hike in to the climb in the approach shoes, then switch to the boots when the real mountaineering starts. There are specialized shoes sold for this, such as the Evolv Cruzer, that flatten down really small in your pack when you're not using them.
