Video footage circulated on social media following the game of one person urinating into a plastic cup before handing to another to throw towards the visiting rival fans at Wembley Stadium.
Spurs investigated the incident and, after identifying the individuals concerned, have now issued lifetime bans to both men.
A Tottenham spokesman said: "This kind of behaviour is not acceptable and we shall be issuing lifetime bans to both individuals in the video."
The north London club had led 2-0 in last Wednesday's tie before the Hammers produced a second-half recovery to take their place in the quarter-finals, where they were drawn away to Arsenal.
It was the second time Tottenham and West Ham have faced each other this season, with Spurs running out 3-2 winners at the London Stadium in the Premier League.
Additional reporting by Press Association.
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