A national newspaper's website picked up on the tweets - sent in 2012 - on the day Robinson was handed his England cap and took two New Zealand wickets as England struggled at Lord's and Robinson issued an apology after the first day of the Test against New Zealdn at Lord's.
Sussex Cricket released a statement on Wednesday. Here it is in full.
"There is simply no place for discrimination of any kind in cricket or anywhere else. We’re committed to making cricket in Sussex a game for everyone, so it goes without saying that we were beyond disappointed to read these tweets when they were brought to our attention today. Their content was wholly unacceptable.
"We are pleased that Ollie has apologised unreservedly and taken responsibility for a significant mistake that he made as a teenager. His age does not excuse the content of these tweets in any way and he will now suffer the consequences of his actions.
"In the years since the tweets were posted, Ollie has matured hugely. The Ollie Robinson we know at Sussex is very different from the young man that sent these tweets. We know he recognises the severity of the situation and that he is devastated that what should have been a proud day has been overshadowed in this manner. We know also that Ollie will learn some very important lessons from this experience. We will be here to offer any support Ollie needs during that process."