A sprained finger is usually not a big deal. Who hasn’t at some time wrenched a finger?
Yet occasionally, what appears to be a sprain might be much more. For example, a small fracture could easily be missed if an xray is not done. this fracture, if displaced might cause later difficulties moving the finger unless it is reduced.
Another problem occurs if the joint is dislocated. All fingers except the thumb, have 2 joints (the thumb only has one). If the finger is swollen enough, it would be easy to miss the dislocated joint. And the joint might be fractured as well, a condition known as fracture dislocation of the joint. Some of these might require surgery.
Tendons can be torn from the injury. There are many tendons which help move the fingers. Some bend the fingers and other extend them. With a sprain, fracture or dislocation, the tendons might be partially or completely torn and some of the pulley systems in place to ensure smooth movement, could be damaged as well. Untreated, these injuries could result in permanent deformities of the fingers such has have a finger stuck in the bent position.
This doesn’t necessarily mean that you need to rush to the ER for every minor finger sprain. If the finger is really swollen, you can’t move it well or the way you injured the finger involved a lot of force, it might be better to make the trip to the local hospital.