Maybe it's helpful to add a longer comment, in community-wiki format. The original question is not well-formulated, I think, as shown in the later convoluted remarks on the case $\theta =1$. It's probably better here to follow Bourbaki (Chapter VI), since the essential problem concerns just an irreducible root system in a vector space $V$ having as basis a fixed choice of simple roots. This choice then determines a partition of the complement in $V$ of reflecting walls for the various roots into Weyl chambers. One of these (call it $C$) is the dominant Weyl chamber (called "positive" in the question) defined by $\langle \lambda, \alpha_i^\vee \rangle >0$ for all simple $\alpha_i$ with $\lambda \in V$. Its closure $\overline{C}$ is then a fundamental domain for the action of the Weyl group $W$.
The fundamental weights $\varpi_i$ lie in the walls of $C$ and are at acute (or right) angles. On the other hand, the simple roots $\alpha_i$ are at obtuse (or right) angles and determine a "positive root cone" (call it $D$) consisting of positive linear combinations of simple roots. Then $D$ contains $C$ because each $\varpi_i$ is a positive $\mathbb{Q}$-linear combination of the $\alpha_i$. But $D$ is usually larger than $C$.
We are given an automorphism $\theta$ of the Dynkin diagram, for example the one of order 2 for type $A_\ell$ when $\ell \geq 2$ (coming from $\mathrm{SL}_{\ell+1}$) which switches $\alpha_i$ and $\alpha_{\ell-i+1}$. The question then concerns $(*) \:\lambda - \theta w \lambda$ for a fixed $w \in W$ and any dominant weight $\lambda$ (say in $\overline{C}$). Bourbaki's Prop. 18 says for $\theta =1$ that $\lambda - w \lambda$ lies in $\overline{D}$, but of course usually not in $\overline{C}$.
First write $\lambda$ as a $\mathbb{Z}^+$-linear combination of the $\varpi_i$, say with coefficients $c_i$. Then $(*)$ is a $\mathbb{Z}$-linear combination of simple roots. Maybe the intended question for arbitrary $\theta$ is whether there is a cone inside $D$ (or $\overline{D}$) consisting of those elements $(*)$ defined by conditions on the $c_i$ such as those in the example. Of course, the cone need not lie in $\overline{C}$, as shown by the case $\theta =1$. Anyway, the version I've stated seems likely to have a positive answer, but I don't know how to prove it.