Is it true that something carved into the side of a tree will stay at the same distance from the ground forever, regardless of how tall the tree grows?

Yup. In your basic tree, you got your primary meristems, which are growth areas located at the tips of the roots (below) and shoots (above), and your secondary meristem (also called the cambium), which is a thin sheath wrapped around the tree between the bark and the wood. The meristems are where all cell division, and thus all growth, occurs. The primaries make the tree taller, the cambium makes it fatter. So when your local sequoia adds a couple stories at attic level, the tree gets taller, but everything at ground level pretty much stays put.

From: The Straight Dope by Cecil Adams