Paul B. Larson mentions a partial result in *A Brief History of Determinacy*, page 48. Sadly he doesn't give a precise source.

Foreman, Magidor and Schindler showed that
if there exist infinitely many cardinals $δ$ above the continuum such that the
tree property holds at $δ$ and at $δ^+$, then [the axiom of projective determinacy] holds. The hypothesis of this statement had been shown consistent relative to the existence of infinitely many supercompact cardinals by James Cummings and Foreman. It is not known whether the conclusion can be strengthened to $\text{AD}^{L(\mathbb{R})}$.

So the tree property for all successors >$\omega_1$ has to be a very strong statement, at least as strong as the existence of infinitely many Woodin cardinals. This is the best partial/related result I know of.

EDIT: See this paper, which has the following abstract:

Starting from a strong cardinal and a measurable cardinal above it, we construct a model of ZFC, in which, for every singular cardinal $δ$, $δ$ is strong limit, $2^δ = δ^{+++}$, and the tree property holds at $δ^{++}$. It answers a question of Friedman, Honzik and Stejskalova. We also produce, relative to the existence of a strong cardinal and two measurable cardinals above it, a model of ZFC in which the tree property holds at all regular even cardinals. The result answers questions of Friedman, Halilovic and Honzik .

Which answers the question from the title, i.e. it is possible to have arbitrarily large cardinals with the tree property (relatively to very large cardinals).