Coffee is an item in House Party. It is located on an end table next to the couch in the Dining Room.
In your inventory, the Coffee/Empty Cup/Hot Coffee looks identical to the Mug, but the two items are not interchangeable.
You can give the Coffee to Patrick, but he initially refuses it because he thinks it will sober him up. You can then suggest that it's Irish coffee to get him to reconsider. This allows you to give it to him again. However, Patrick will not drink the Coffee if it isn't hot.
When the Coffee is used with the Kettle, the Coffee becomes an Empty Cup. After you have heated the Kettle, using it with the Empty Cup turns the Empty Cup into Hot Coffee. Patrick will take the Hot Coffee, allowing you to complete Fuck Tuesday.
You can also use the Coffee with Dusty Penguin in Artsy Fartsy, though you will lose the art contest.