Sadly, running shoes won't last forever. As a general rule of thumb, most running shoes need replacing after 3-500 miles, dependent on how intensely you use them, how much you weigh, and how well you look after the shoes. To clarify, your shoes won't fall apart after 300 miles; it's more to do with the lifecycle of the foam – after a while, it will return less energy, which might compromise running form and make you run slower.
Some people will experience the upper giving in first – depending on where the holes appear, you might want to consider changing to a larger size (if the toes poke through the front of the upper), or picking shoes that are better suited for wide feet (the lateral side of the upper frays). If your heels rub, you might have a narrow feet; you should consider sizing up and checking for narrow-fit shoes.