No. Aside from the fact that Tesla was over 80 years old in 1939 (which would have made him by far the oldest person on the project — by comparison, Niels Bohr was only in his 50s and was considered pretty old, and Richard Tolman, who assisted General Groves, was considered ancient at 61), Tesla was by that point a shadow of his former inventive self and considered more or less a crank. He actively refused to participate in the physics revolutions of the decades prior (relativity and quantum physics, both of which he denied), and would not have for a moment been seen as someone to involve with such an endeavor. He was by all accounts living a sad life in New York City hotels, feeding (and talking to) pigeons, occasionally commenting on world and scientific affairs to journalists, but not at all someone considered scientifically active.

From a technical standpoint, the nuclear program would not likely have been very useful for someone of Tesla's earlier standing. One can imagine an alternate universe in which a younger Tesla participated in the development of radar, for example — that is more up his alley, technically. But there is little on the Manhattan Project that bears much on the topics that Tesla himself worked on.