When the Sole Survivor first enters Goodneighbor, Finn will approach and demand the player character pay an "insurance fee," otherwise "accidents" may happen to them. The Sole Survivor can choose to threaten Finn to back off or kill him.
Whatever option is chosen, Finn will always end up dead: if the player character doesn't kill him, then mayor John Hancock will appear and demand Finn to let them go. Finn will refuse and subsequently have a quarrel with Hancock in which he states "one day there'll be a new mayor". This appears to be a poor choice of words, as it results in Finn being stabbed to death by Hancock.
If one kills Finn with a switchblade, they can end up stabbing Finn to death in a similar manner to the way Hancock kills him.
If the player talks to Finn with Nick Valentine as a companion, the conversation will be slightly different but end in the same result. Hancock will also berate Finn for harassing the Sole Survivor and Nick during one of his "rare visits."