近寄る and 近づく are closer in meaning than 迫る.
近寄る means quite literally to approach. It can also meaning to get close to someone emotionally. Because 寄る means to come or approach physically, its uses basically stop there. On a personal level, I've been competent in Japanese for 10 years and I've only ever seen this word written, not spoken. Even written it's not that common.
近づく has several more layers however. While also meaning to approach physically as well as emotionally, it is also used to describe approaching dates and events. It can also be used to describe approaching an ideal certain circumstance.
迫る on the other hand is used to describe something approaching in an impending sense. This can be used to describe something physical (an army, for example) or otherwise (dates, events, calamity, etc.).
Definitions in Japanese can be found here: