If I temporarily remove an external HD from my Mac, then re-introduce it a few weeks later, will Time Machine back it all up again?
If this is a Time Machine disk, disconnecting it and then reconnecting it in the future will simply make Time Machine update it when it is plugged in as normal, meaning it will back up whatever has changed. It will not erase the drive and start from scratch. It will simply resume where it left off.

Or do you mean that this external disk was one of the drives that Time Machine was backing up onto another, separate drive? If so, it seems that if you use that drive on another computer and then try to back it up with Time Machine again, it may cause some problems (or may not. Just depends.) Also, in a case where TM hasn't backed up a drive in a while, it may lose much of its cache for that disk and appear to be backing everything up again when you plug it back in. However, in this case it seems that it's not actually backing it all up, it's only really keeping what has changed.