Drupal Association members fund grants that make connections all over the world.
I'd like to suggest not displaying the Tweet button on nodes that are being built to be printed. Since there's no way for a user to interact with the button on the printed page, it doesn't seem important to build it and add it to the node.
The quickest way to implement this would be a test for
$node->build_mode == NODE_BUILD_PRINT before adding the button to $node->content in tweetbutton_nodeapi.
It also seems to me that it would be more thorough to only add the button to nodes if the build mode is regular and possibly preview (NODE_BUILD_NORMAL and NODE_BUILD_PREVIEW, respectively). I suspect it's not terribly important that the Tweet button get indexed for search, for example.
Thanks for tweetbutton!
 There's a list of node build modes in node.module