Let $L$ be a simple Lie algebra of Cartan type of absolute toral rank 2 over an algebraically closed field $\mathbb{F}$ of characteristic $p\geq 5$.
Denote by $L\_{[p]} $ the minimal $p$-envelope of $L$ and let ${\mathfrak t}$ be a 2-dimensional torus of $L\_{[p]}$.
For a nontrivial irreducible restricted $L\_{[p]}$-module $V$ we denote by $\Gamma(V, {\mathfrak t})$ the set of weights of $V$ with respect to ${\mathfrak t}$.


In Section 10.7 of the book [H. Strade: Simple Lie algebras over fields of positive characteristic. II. Classifying the absolute toral rank two case] it is showed that $\Gamma(V, {\mathfrak t}) \cup$ {0} is a 2-dimensional vector space over the prime field $\mathbb{F}\_p$. 


In some cases the 0 weight is actually missing in $V$. For instance, this happens when $L$ is the Hamiltonian algebra $H(2;\underline{1};\Phi(\tau))^{(1)}$ and $V$ is the adjoint module. On the other hand, in this case does there exist a nontrivial restricted irreducible module $V$ for $L\_{[p]}$ such that the 0 weight is in $\Gamma(V, {\mathfrak t})$? 



Weights of restricted modules of some Cartan type Lie algebras 
Such modules do exist for $p>3$. The Lie algebra $L\_p={\rm Der}\ H(2;\underline{1};\Phi(\tau))^{(1)}$ has a filtration of depth $1$ such that ${\rm gr}(L)$ is a graded Lie algebra of type $\rm H$. This can be seen by embedding it into $W(2;\underline{1})$ as explained in Strade's paper in Can. J. Math. , vol. 43, 1991, 580-616. As $TR(L)=2$ the Lie algebra $L\_p$ contains a $2$-dimensional torus, $T$ say. Any $2$-dimensional torus of $L\_p$ must intersect trivially with $(L\_p)\_{(0)}$ because any nonzero torus of $(L\_p)\_{(0)}$ is $1$-dimensional and has no zero weight on 
$L\_p/(L\_p)\_{(0)}$ (here $(L\_p)\_{(0)}$ is the zero component of the filtration).
Passing to corresponding graded algebras and modules and using some results on representations of $H(2;\underline{1})^{(2)}$ mentioned in Jim's answer one can see that
all but two of the irreducible restricted representations of $L\_p$ are induced from irreducible restricted representations of $(L\_p)\_{(0)}$. The equality $L\_p=T\oplus (L\_p)\_{(0)}$ then shows that all possible $T$-weights (including zero) occur in such modules with the same multiplicity. 


As a historical remark I should add that irreducible representations of finite dimensional graded Cartan type Lie algebras of type ${\rm W, S,H}$ were first studied in the late 70s by Ya. Krylyuk, a former PhD student of Kostrikin. Unfortunately, the results of his PhD thesis were buried in the cemetery of "secondary scientific and technical information" called VINITI and remained completely unknown to the rest of the world (the was no ${\rm arXiv}$ at the time). Wikipedia now has a comprehensive article about VINITI.


