In a break from decades of tradition, the charity said it will pull out of attending events to focus on directing funds to other parts of its operations such as crisis support.
It came as the charity was called to assist at the industrial estate where 39 bodies were discovered in the back of a lorry yesterday morning.
Michael Adamson, chief executive of British Red Cross, said: "In recent years, we have seen increasing pressures on both our income and the demands for our assistance, which means we must prioritise how we use every pound donated to us.
"Unfortunately, our event first aid work has been running at a financial loss for some time.
"The service still requires £1.8m of donations annually to cover the shortfall between income and costs - and this is diverting vital funds from our efforts to provide emergency support for major domestic and global crises.
"So it is with real sadness that we have taken the very difficult decision to close our event first aid service by 31 March 2020.
"This is not a decision our trustees have taken lightly.
"We sincerely hope our volunteers will consider continuing their journey with us in other ways."
Mr Adamson said the Red Cross was working with St John Ambulance and St Andrew's First Aid to transfer volunteers who wish to continue offering first aid.
Around 55 staff are expected to be involved in a consultation, with some potentially moved into other departments.
Vehicles from the British Red Cross attended the scene of the Essex lorry disaster at the request of Thurrock Council yesterday.
The charity said its staff and volunteers were providing emotional support to members of the emergency services and others dealing with the tragedy.