For context, Tesla was 79 at the time he gave this interview. He had spent his entire life working on electromagnetism within the framework of prerelativistic theories, such as aether theories and a concept according to which "all space is filled with a gaseous substance." He accomplished some important things, but he was also somewhat of a kook and a charlatan. He had only two years of formal university education. In the New York Times interview referred to in the question, he promised an apparatus, which he never actually produced, "by which mechanical energy can be transmitted to any part of the terrestrial globe."
How can this be? If Einstein's theory is falsifiable yet very reinforced by evidence (the balance for a good scientific theory) why did Tesla disavowed it so strongly?
In 1935, there was actually not very much direct, strong evidence supporting relativity. Evidence like Michelson-Morley and Eddington's 1919 eclipse observation could have been considered amiguous or uncertain. Ives-Stilwell wouldn't happen until 1938, Rossi-Hall 1941.
Tesla actually claimed to possess evidence that falsified special relativity. The same 1935 interview says: "He has measured cosmic ray velocities from Antarus ... which he found to be fifty times greater than the speed of light..."
In addition, Tesla claimed to have simple thought experiments that disproved relativity, and he disliked relativity on aesthetic, philosophical, and religious grounds:
Tesla contradicts a part of the relativity theory emphatically, holding that mass is unalterable; otherwise, energy could be produced from nothing, since the kinetic energy acquired in the fall of a body would be greater than that necessary to lift it at a small velocity.
-Hugo Gernsback, Science and Mechanics, November 1931
What is 'thought' in relativity, for example, is not science, but some kind of metaphysics based on abstract mathematical principles and conceptions which will be forever incomprehensible to beings like ourselves whose whole knowledge is derived from a three-dimensional world.
I hold that space cannot be curved, for the simple reason that it can have no properties. It might as well be said that God has properties.
New York Herald Tribune, 1932 Sep 11
We read a great deal about matter being changed into force and force being changed into matter by the cosmic rays. This is absurd. It is the same as saying that the body can be changed into the mind, and the mind into the body. We know that the mind is a functioning of the body, and in the same manner force is a function of matter. Without the body there can be no mind, without matter there can be no force.