On page 393 of this paper, Speyer and Sturmfels produce a quotient of a particular geometric object (which happens to be a polyhedral fan), and claim that, as in the question title, "no cone in this fan contains a non-zero linear space".
How would one go about showing something like this?
The fan in question is a quotient of the Tropical Grassmannian $\mathscr G_{2,n}'$:
$$\mathscr G_{2,n} = \bigcap_{i<j<k<\ell}H(p_{ijk\ell})$$ where $H$ denotes the tropical roots of a tropical polynomial (the points where the minimum is achieved twice), and $p_{ijk\ell}$ is a tropical polynomial in variables indexed by 2-sets of $\{1,\dots, n\}$: $$p_{ijk\ell} = \min\left(X_{ij}+X_{k\ell},~ X_{ik}+X_{j\ell},~ X_{i\ell}+X_{ jk}\right).$$ Then $\mathscr G_{2,n}'$ is the image of $\mathscr G_{2,n}$ under quotienting by the image of $\varphi:\Bbb R^n\to\Bbb R^{\binom n2}$ where $\varphi(e_i)=\sum_{j\neq i} e_{ij}$.
but I am interested in producing a similar statement about other fans, so general techniques are most useful. However, S&S don't bother to prove this statement anywhere and it seems like something quite non-obvious even in this example. So I have nothing to work from, and I would be interested in seeing any concrete nontrivial worked example.