ITV said the Pakistan-born actor, 45, who joined the soap as Sharif Nazir in February 2014, would not be returning to the show “with immediate effect”.
Screenshots of messages posted on his private Twitter account appear to hit out at Indians over the occupation of Kashmir.
They include “offensive language” used to refer to Indian people.
The screenshots, posted first by the Sunday Mirror, also call for Pakistani actors to stop working in India.
The alleged comments come after the Indian army said it killed a freedom fighter in a gunfight in the occupied region of Kashmir, which is claimed by both India and Pakistan.
Mr Anwar’s character had been a member of the first Muslim family on Coronation Street.
In a statement, ITV said: “We are deeply shocked by the entirely unacceptable, racially offensive comments made on Twitter by Marc Anwar.
“We have talked to Marc and, as a consequence of his comments, he will not be returning to Coronation Street with immediate effect.”
ITV confirmed that pre-filmed episodes with the actor, who has also appeared in Hollywood films such as Captain Phillips and 51st State, will still air but his presence in the show will be minimised until his character can be written out.