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 $\mathbb{Q}$-linear combination of simple roots. Maybe the intended question for arbitrary $\theta$ is whether there is a cone inside the $\mathbb{Q^+}$-span of $D$ (or $\overline{D}$) consisting of those elements $(*)$ defined by conditions on the $c_i$ such as those in the example. (A fixed denominator may occur, coming from the index of the root lattice in the weight lattice.) 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.