Some results supplementing Joel's answer:
- Shelah proved (around 1995) that if $\lambda$ has cofinality $\omega$ and is the supremum of strongly compact cardinals, then $\lambda^+$ has the tree property. See
Menachem Magidor, and Saharon Shelah. The tree property at successors of singular cardinals, Archive for Math Logic, 35 (5-6), (1996), 385-404. MR1420265 (97j:03093).
- Neeman proved that, assuming the existence of $\omega$ supercompact cardinals, we can force a model where the tree property holds at all the $\aleph_n$ ($2\le n<\omega$) and at $\aleph_{\omega+1}$. See
Itay Neeman. The tree property up to $\aleph_{\omega+1}$, preprint.
Neeman's result improves previous results, both in terms of the cardinals with the tree property, and in consistency strength: Magidor and Shelah had obtained the tree property at $\aleph_{\omega+1}$ from a huge cardinal with $\omega$ supercompact cardinals above. As mentioned in Joel's answer, Cummings and Foreman had obtained the tree property for the $\aleph_n$ ($2\le n<\omega$), also from $\omega$ supercompact cardinals. At the moment, Neeman's is the best current result in terms of intervals of regular cardinals with the tree property. At least in $\mathsf{ZFC}$.
- Arthur Apter proved (around 2009) that the following is consistent, relative to a proper class of supercompact cardinals: $\mathsf{ZF} + \mathsf{DC} +$ Every successor cardinal is regular and has the tree property, while every limit cardinal is singular. See these slides, and
Arthur W. Apter. A remark on the tree property in a choiceless context, Arch. Math. Logic, 50 (5-6), (2011), 585–590. MR2805298 (2012d:03115).
The conclusion of Apter's result implies determinacy in $L(\mathbb R)$, and more.
- The upper bound in consistency strength for successive cardinals with the tree property is a supercompact cardinal with a weakly compact cardinal above it. Around 1983, Abraham forced, from these assumptions, that $2^{\aleph_0}=\aleph_2$, and both $\aleph_2$ and $\aleph_3$ have the tree property. All results on successive cardinals with the tree property build on Abraham's argument. See
Uri Abraham. Aronszajn trees on $\aleph_2$ and $\aleph_3$, Ann. Pure Appl. Logic, 24 (3), (1983), 213–230. MR0717829 (85d:03100).
- The best known lower bound is due to Foreman, Magidor, and Schindler. They show that if all $\aleph_n$ ($2\le n<\omega$) have the tree property, and $\aleph_\omega$ is strong limit, then $\mathsf{PD}$ holds. See
Matthew Foreman, Menachem Magidor, and Ralf Schindler. The consistency strength of successive cardinals with the tree property, J. Symbolic Logic, 66 (4),(2001), 1837–1847. MR1877026 (2003m:03083).
This result is frustrating in the sense that we expect two successive cardinals with the tree property should give us much more in consistency strength than this, beyond $\mathsf{AD}^{L(\mathbb R)}$, and likely beyond the current reach of descriptive inner model theory. Still, this would be frustratingly short of the best current upper bounds, which experts expect are much closer to the truth.