Instead a clip of the 51-second song will be aired as part of a Newsbeat report, BBC Radio 1 controller Ben Cooper has said.
Sales of Ding Dong! The Witch is Dead have soared since Margaret Thatcher’s death on Monday, aged 87.
Ben Cooper called the decision “a difficult compromise”.
The song is set to take No 3 spot in Sunday’s countdown, according to the Official Charts Company.
Speaking to BBC Radio 1 Newsbeat, Ben Cooper said: “The decision I have made is I am not going to play it in full but that I will play a clip of it in a news environment.
“When I say a news environment, that is a newsreader telling you about the fact that this record has reached a certain place in the chart and here is a clip of that track.
“It is a compromise and it is a difficult compromise to come to. You have very difficult and emotional arguments on both sides of the fence.
“Let’s not forget you also have a family that is grieving for a loved one who is yet to be buried.”
The announcement was followed by a statement from the corporation, reading: “The BBC finds this campaign distasteful but does not believe the record should be banned.”
“On Sunday, the Radio 1 Chart Show will contain a news item explaining why the song is in the charts during which a short clip will be played as it has been in some of our news programmes.”
The original song was performed in the 1939 Judy Garland film by characters celebrating the demise of the much-hated Wicked Witch of the East.
The campaign has been called “tasteless” by the BBC’s director general Tony Hall, Deputy PM Nick Clegg and others, though few have called for the song to be banned outright.
DJ Paul Gambaccini and former director general Greg Dyke are among those who have taken the opposite position, arguing that the chart is a statistical reality that should not be editorialized.
Ding Dong! The Witch Is Dead, attributed to Judy Garland and the cast of the movie, is also expected to feature in the Capital Chart’s Big Top 40.
A spokesman for Capital FM said the station was currently “reviewing the situation”.
Rival campaigns are under way to get a song considered to be more favorable to Margaret Thatcher into this week’s countdown as well.
One Facebook group is seeking to boost sales of (I’m In Love With) Margaret Thatcher, a 1980 track by the punk band Notsensible.
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