Sneakers do not protect your feet from sharp rocks. Without a stiff footbed they will get battered and possibly injured if you aren't careful about where and how you step, especially with a heavy pack.
They have no ankle support. The high top and, more importantly, sturdy heel cup and stiff sole of a hiking boot offer better resistance to rolling your ankle.
Most sneaker rubber grips poorly to rock, and the soft soles are terrible for edging. They are probably the worst footwear for technical rock climbing.
The low tops let in snow, dirt, and mud, and leave your ankles exposed to talus. The latter is probably the only considerable safety concern regarding sneakers in the mountains.
They are light, greatly reducing fatigue in many situations.
They usually dry somewhat quickly, meaning you can often wear them on a fording without worrys about cold feet days later.
Personally, I use trail runners for nearly all hiking where I don't anticipate significant technical climbing, snow, or talus, and where my load is light (30lb or less). In the northeast US, that's most rock approaches. Out west and in Canada, it's much rarer. But, you may be able to hike for a long way before encountering any of the first three - if so, sneakers will be great.