"I am disgusted by this so-called Chelsea fan," Lampard said at Friday's press conference.
"I don’t know how it’s allowed - it’s too easy. Something needs to be done, as well as obviously changing mindsets completely.
"I’m so angry for Tammy and angry for us as a club, because that’s not what we’re about. The club does a lot of work against discrimination at all levels and it’s a setback when these things happen.
"Obviously he is more than disappointed - who wouldn’t be? Tammy asked me to take the fifth penalty because he wanted to stand up on a big night when the world was watching.
"Moments afterwards, somebody sitting behind a keyboard or a phone has said the most disgusting things possible.
"He is always desperate to score and that’s one of the things I love about him. He gets in positions regularly and that’s an indication of his character that he’s not hiding."
Chelsea produced an improved display in Istanbul, three days on from suffering a 4-0 loss to Manchester United in their opening Premier League match.
Lampard felt his side should have won the all-English affair in Turkey and is now looking to get off the mark with his first victory in Sunday's home match against Leicester City.
"We didn’t concede so many goals," Lampard said. "There were some really good parts in the Manchester United game. We just need to learn about the four errors from the goals.
"Liverpool are one of the best teams in the world and we matched them. We moved the ball and created chances.
"We lost on penalties, but in the game we should have won. What’s important now is the reaction and we’re looking to get our first three points.
"We could have done without extra time, but it is what it is. We have an extra day until Sunday and the players are doing a lot of recovery rather than training."
With the transfer window for European clubs open until early next month, a number of Chelsea's fringe players have been tipped to move on in the next couple of weeks.
Lampard has ruled out an exit for Michy Batshuayi, who is behind Olivier Giroud and Abraham in the pecking order, but the futures of right-back Davide Zappacosta and midfielder Tiemoue Bakayoko are less clear.
"Michy will stay here as one of our strikers," Lampard said. "We have three strikers this season and there will be opportunities for him. He’s very much our player and competing with Tammy and Ollie.
"As for Zappacosta, that is not clear yet. He is our player and Reece James is probably out for three more weeks. That will be a decision we have to make before the European transfer window closes, similar to Bakayoko."