can hammering a penny into a tree kill it over time?

Answer 1

It's possible, especially if the tree is young or if the coin has contaminates on it.

There's been a practice of hammering coins (mainly pennies or 2 cent quids) into FALLEN trees, trunks, or logs going way back. Some believe it was a common practice similar to throwing coins in wells - for wishes- & called them "money trees" or "Wish Trees. Others say it is for safe passage. Romans would offer a coin to the "Gods" before crossing water.
Here's some pictures & the rest of the story:
http://community.livejournal.com/faeid/

Answer 2

Possibly the wound will allow organisms to get into the tree and infect it. More likely the tree will simply grow scar tissue around the coin.

Girdling the tree (remove a 3" wide strip of bark from all around the circumference of the tree just a few inches above ground level) will kill it within a year or two. Then you will have a dangerous standing dead tree.

If you want to remove a tree, best to cut it down and treat the stump with Round Up.