I am using this jQueryUI tooltip.
The jQueryUI file is quite large as it contains all the other elements.
Do I need all that code to let the tooltip behave properly or could I remove some of the code?
No, you don't.
Here's your jQuery UI build with only tooltip selected: http://jqueryui.com/download/#!version=1.10.3&components=1101000000000000000100000000000000
Feel free to customize this selection if you want.