Can a hairline fracture in the hand get worse if left untreated?

Yes but . It really depends a lot on location ie a scaphoid fracture that leads to avascular necrosis or nonunion is not good but a matacarpal fracture may just hurt until teated, then get better quickly.
Yes. Yes, "hairline fractures" can get worse. Whether your fracture is from trauma or a stress fracture you must remember that the fracture is a weak place in the bone. If ample force is applied to the area the fracture could become larger or displace. Also, using an injured hand puts you at risk of injury to other areas that may come from improper mechanics when you compensate due to pain/ weakness.