I usually just run 1 day per week, I just do my long run for 20k, but now I have a minor injury in my finger, I have a mallet finger (or baseball finger) so I need to use a stack split in my finger for 6 weeks to recover. So I don't know if the recovering of the long run can affect the recover of my finger.
Ask yourself this question:
What's the likelihood of re-injuring the finger?
The finger will take as long as it takes to get better, and running isn't going to change that significantly enough to worry about it if at all. As long as there is a low likelihood that you are going to hurt the finger again doing any activity, there is no reason to avoid it.
You may experience minor pain from the impact of your feet radiating up to your finger, or you may not. I don't think it's re-injuring it, but it might not feel good. A bit of ice for 20 minutes will probably make it feel better after your run.
You will also probably have to be careful if you are grabbing some water with that hand while running.
Those are the only two things I can think of that would cause you any problems.