The International Equestrian Federation (FEI) also suspended the 32-year-old Lynch from competing in the sport for three months.
The ban runs out November 21 after taking effect from the day Lynch was provisionally suspended from the Games.
His horse Lantinus tested positive for capsaicin, a banned pain relieving medication derived from chili peppers.
Lynch and Lantinus finished tied for seventh in qualifying in the individual jumping class but were prevented from taking part in the final, which was held in Hong Kong.
The world governing body ordered him to pay 3,750 Swiss francs (£1,908) as a fine and legal costs.
Lynch has 30 days to appeal the verdict at the Court of Arbitration for Sport but he said that he will not be challenging it.
He said: “With the FEI taking nearly two months to announce its decision, there seems little point me now appealing it.
“I just want to get back in the ring and I feel the best way that I can thank my family, friends and my owner, Thomas Fruhmann, for their support over the past number of weeks is to start winning again.
“Before I do, however, I will go back home to thank people personally.”
It is the second consecutive Olympics that an Irish show jumper has been disqualified and banned.
In Athens four years ago Cian O’Connor was stripped of the gold medal after his horse Waterford Crystal tested positive.
Joe Walsh the chairman of Horse Sport Ireland, said that the group would study the FEI ruling to ensure it never happened again.
He said: “I want to assure people that we will be relentless in dealing with this issue.”
Five horses failed dope tests at the Games.