There are a couple of pieces of NBN ICE that start with non-subroutine text stating something along the lines of, "When this ICE is encountered, do this and this and this..."

There is a Criminal card that tells the runner to make a run and ignore or bypass the first piece of ICE encountered during that run.

So, does the criminal player still have to contend with the starting text on the ICE card, or is the entire thing ignored?

It's ignored. You can find the relevant ruling in the official FAQ.

Boom, thanks!