Is it possible to have the iPhone automatically open iTunes/sync on one computer and not on others?
iTunes/iPhone syncing is done on a "per Mac" basis. In iTunes Preferences, go to Devices, and check the box for "Prevent iPods, iPhones, and iPads from syncing automatically". This will prevent iTunes from opening when a device is plugged in.

UPDATE: There used to be in previous versions of iTunes, in the old "Syncing" Preferences, a way to disable automatic syncing and that prevented iTunes from even opening. But it seems they have changed the behavior because searching for answers on other sites yield the same behavior but in previous versions of iTunes. I set this preference ages ago, so the preference must be holding over.

In response to your comments:


  iTunes still starts, still does some
  kind of mini-sync, but doesn't
  recognise the name of the phone or
  allow access to its contents.


Are you sure you are syncing with the iTunes install you originally started with? Syncing is strictly a one-to-one relationship for iPhones and iTunes. Once you sync an iPhone to an iTunes install and wish to sync with a different one (like on another Mac), you are then forced to clear the phone and start anew. 

My iTunes behaves the way you are describing, but when I plug in my iPhone, I get the "mini-sync" you mention because that's the device registering with the OS and iTunes that it exists and iTunes puling needed info to interact with it. Can't really get around that one.


  it would prevent the machine being
  used at all for syncing, no?


No. You could still sync if you wanted, but that's if you are syncing with the iTunes install that recognizes the phone. Otherwise, see my comment above.

BTW, save the downvoting for the really egregious answers. Downvoting costs points. Otherwise, just leave it alone.