I have been told that Kurt Godel was among the Mathematicians that Einstein consulted on occasion in his work. Can anyone tell me if this is true or not, or if it is at least plausible? If it is true, would that be an example of a Philosopher making a contribution to Physics?
I do not think "consult" is the right word. They worked together at the Institute of Advanced Study in Princeton, became personal friends, their offices were nearby and they had many conversations. Todorov tells the story of their friendship in Kurt Goedel and His Universe, see also Berlinsky's Einstein and Gödel. But that only happened after 1940 when Goedel finally emigrated from Vienna. By that time Einstein's major work, special and general relativity, quantum statistics and entanglement, etc., was already done. As Morgenstern reminisced in a 1965 letter to Bruno Kreisky,
"Einstein told me that his own work no longer meant much, that he came to the Institute merelyum das Privileg zu haben, mit Gödel zu Fuss nach Hause gehen zu dürfen (to have the privilege to be able to walk home with Gödel)".
Gödel, on the other hand, was still active, but it is unclear that he consulted Einstein either, even when he ventured in 1948 into constructing a solution of general relativity with closed timelike curves, the well-known Gödel's universe. Todorov adds:
"Einstein is such a legend that most people are afraid to approach him. Gödel isn't. There is a feeling of equality between them. Their debates range from the trivial to the profound. Gödel is skeptical about Einstein's idea of a unified field theory and says so ([Y02] p. 56). [...] In spite of being close to each other it is not clear whether Gödel discussed his work on general relativity with Einstein prior to its completion. As witnessed by Straus, Gödel was totally solitary and would never talk to anybody while working."
As for a philosopher contributing to physics, both Gödel and Einstein were part-time philosophers. The ideas of relativity came, in part, from Einstein's reflections on the positivist philosophy of Mach. His defense of metaphysical realism in debates with Bohr over completeness of quantum mechanics are also well-known. But as Einstein was first and foremost a physicist, so Gödel was first and foremost a mathematician, a mathematical logician more precisely, and only then a platonist philosopher. And that was how Einstein saw it. According to Strauss:
"Einstein... felt that he should not become a mathematician because the wealth of interesting and attractive problems was so great that you could get lost in it. In physics he could see what the important problems were and could, by strength of character and stubbornness, pursue them. But he told me once 'Now that I've met Gödel, I know that the same thing does exist in mathematics'".