I think it's more likely at the Drama Desk because they have around 130 voters, while there are about 868 voters for the Tonys. A tie would be far more difficult with a much larger group of voters.
There have been 10 Tony ties in history, the last being in 2009 with Billy Elliot and Next to Normal for orchestrations.
Actress in a Musical (195

: Gwen Verdon, New Girl in Town and Thelma Ritter, New Girl in Town
Featured Actor in a Musical (1959): Russell Nype, Goldilocks, and Leonard Stone, Redhead
Musical (1960): The Sound of Music and Fiorello!
Actress in a Musical (1962): Anna Maria Alberghetti, Carnival! and Diahann Carroll, No Strings
Actress in a Musical (196

: Patricia Routledge, Darling of the Day and Leslie Uggams, Hallelujah, Baby!
Costume Design (1977): Theoni V. Aldredge, Annie and Santo Loquasto, The Cherry Orchard
Actress in a Play (1979): Constance Cummings, Wings and Carole Shelley, The Elephant Man
Scenic Design (1980): John Lee Beatty, Talley's Folly and David Mitchell, Barnum
Score (1993): John Kander/Fred Ebb, Kiss of the Spider Woman and Pete Townshend, The Who's Tommy
Orchestrations (2009): Martin Koch, Billy Elliot, The Musical and Michael Starobin and Tom Kitt, Next to Normal