The campaign, booked with JCDecaux Airport's experiential division Sphere and devised by communication agency Circle IMC, extends Ballantine's "Leave an Impression" campaign and claims to be the first of its kind at a UK airport.
Through the work, Pernod Ricard hopes to increase awareness of Ballantine's Finest by word of mouth.
Beever finishes his drawing today (18 December), after which it will be opened to the public until Christmas.
Updates on Terminal 3's digital network will showcase the development of the drawing from its initial stages through to the final piece.
In addition to driving customers to the World Duty Free store in T3, coupons will be handed out offering the chance to win a Ballantine's branded watch.
Travellers will also have the opportunity to sample the Ballantine's range, while a gift is available with any Ballantine's reference purchased.
Nicola Greene, head of Sphere Sales at JCDecaux Airport, said: "A campaign with this level of creative flair demonstrates yet again the potential for experiential advertising at airports.
"We are really proud to be able to deliver a passenger experience that will delight travellers during their airport journey and that can only increase the brand perception of exclusive products like Ballantine's whisky."
Before the launch in London, the Ballantine's Finest Street Impression campaign was first executed in Uruguay outside a shopping mall in Montevideo, followed by Las Condes in Chile in October.