Existence of Aronszajn tree There are theories that prove the existence and non-existence of Souslin trees [exist if $V=L$, don't exist if $\mathsf{MA}(\aleph_1)$] and Kurepa trees [exist if $V=L$, don't exist by Lévy collapsing a strongly inaccessible cardinal to $\aleph_2$].
The existence of Aronszajn trees is provable in $\mathsf{ZFC}$. Is there a theory (stronger than $\mathsf{ZF}$) that refutes the existence of Aronszajn trees? 
 A: Sure. 
If you collapse a weakly compact to $\omega_1$, then the Solovay model you get is one where there are no Aronszajn trees on $\omega_1$. The same would work if you collapse a Mahlo, you'd get that there are no special Aronszajn trees.
This is easy to see in the following argument: If $A$ is a set of ordinals in the Solovay model, then it is definable from a real. Namely, it was introduced by a bounded collapse. But if $T$ is a tree on $\omega_1$, it can be coded by a set of ordinals, and thus was added by a bounded collapse. In the model given by only forcing with that bounded part, the weakly compact we collapse is still weakly compact, so $T$ has a branch there, and therefore in the Solovay model.

Of course, one has to be careful as to what is an Aronszajn tree in this context. Does it have to be well-ordered? Maybe just its initial segments need to be well-ordered? Maybe you want to talk about other cardinals, other than $\omega_1$ itself, in which case the answer is also positive: 

Apter, Arthur W., A remark on the tree property in a choiceless context, Arch. Math. Logic 50, No. 5-6, 585-590 (2011). ZBL1221.03049.

