Maybe I can answer the original question more directly, leaving aside the interesting recent geometric work discussed further in later posts like the Feb 10 one by Chuck: analogues of Beilinson-Bernstein localization on flag varieties and consequences for algebraic groups (Bezrukavnikov, Mirkovic, Rumynin).
The 1979 conference paper by Haboush may be hard to access and also hard to read in detail, but it raises some interesting questions especially about centers of certain hyperalgebras. I tried to give an overview in Math Reviews: MR582073 (82a:20049) 20G05 (14L40 17B40)
Haboush,W. J.,
Central differential operators on split semisimple groups over fields of positive
characteristic.
Séminaire d’Algèbre Paul Dubreil et Marie-Paule Malliavin, 32ème
année (Paris, 1979), pp.
35–85, Lecture Notes in Math., 795, Springer, Berlin, 1980.
The hyperalgebra here is the Hopf algebra dual of the algebra of regular functions on a simply connected semisimple algebraic group $G$ over an algebraically closed field of characteristic $p$, later treated in considerable depth by Jantzen in his 1987 Academic Press book Representations of Algebraic Groups (revised edition, AMS, 2003). After the paper by Haboush, for example, Donkin finished the determination of all blocks of the hyperalgebra.
While the irreducible (rational) representations are all finite dimensional and have dominant integral highest weights (Chevalley), the module category involves locally finite modules such as the infinite dimensional injective hulls (but no projective covers). The role of the finite Weyl group is now played by an affine Weyl group relative to $p$ (of Langlands dual type) with translations by $p$ times the root lattice. In fact, higher powers of $p$ make life even more complicated.
The older work of Curtis-Steinberg reduces the study of irreducibles to the finitely many "restricted" ones for the Lie algebra $\mathfrak{g}$. For these and other small enough weights, Lusztig's 1979-80 conjectures provide the best
hope for an analogue of Kazhdan-Lusztig conjectures when $p>h$ (the Coxeter number). The recent work applies for $p$ big enough": Andersen-Jantzen-Soergel, BMR, Fiebig.
Anyway, the hyperalgebra involves rational representations of $G$ including restricted representations of $\mathfrak{g}$, while the usual enveloping algebra of the Lie algebra involves all its representations. But the irreducible ones are finite dimensional. I surveyed what was known then in a 1998 AMS Bulletin paper. Lusztig's 1997-1999 conjectures promised more insight into the non-restricted irreducibles and are now proved for large enough $p$ in a preprint by Bezrukavnikov-Mirkovic. This and their earlier work with Rumynin use a version of "differential operators" on a flag variety starting with the usual rather than divided-power (hyperalgebra) version of the universal enveloping algebra of $\mathfrak{g}$.
To make a very long story shorter, Haboush was mainly looking for the center of the hyperalgebra (still an elusive beast unlike the classical enveloping algebra center, due to the influence of all powers of $p$). His weaker version of Verma modules may or may not lead further. But there is no likely analogue of the BGG category for the hyperalgebra in any case. That category depended too strongly on finiteness conditions and well-behaved central characters.
ADDED: It is a long story, but my current viewpoint is that the characteristic $p$ theory for both $G$ and $\mathfrak{g}$ (intersecting in the crucial zone of restricted representations of $\mathfrak{g}$) is essentially finite dimensional and requires deep geometry to resolve. True, the injective hulls of the simple $G$-modules with a highest weight are naturally defined and infinite dimensional (though locally finite), but the hope is that they will all be direct limits of finite dimensional
injective hulls for (the hyperalgebras of) Frobenius kernels relative to powers of $p$. Shown so far for $p \geq 2h-2$ (Ballard, Jantzen, Donkin). In particular, the universal highest weight property of Verma modules in the BGG category (and others) is mostly replaced in characteristic $p$ by Weyl modules (a simple consequence of Kempf vanishing observed by me and codified by Jantzen). Then the problems begin, as Lusztig's conjectures have shown. The
Lie algebra case gets into other interesting territory for non-restricted modules.
Best Answer
$\newcommand{\Hom}{\operatorname{Hom}}$Here is an elaboration of my comment with what happens in characteristic $2$ when $d=3$:
The usual decomposition rule gives us a filtration of $S^dV\otimes V$ with two factors: $H^0(4)$ and $H^0(2)$ (I use the notation from Jantzen's Representations of Algebraic Groups and write all weights in terms of the fundamental weight).
Applying the Jantzen sum formula, we see that $H^0(4)$ has a composition series consisting of $L(4)$, $L(2)$ and $L(0)$.
We also see that $H^0(2)$ has a composition series consisting of $L(2)$ and $L(0)$.
All this gives us a composition series $$0 \subseteq M_1 \subseteq M_2 \subseteq M_3 \subseteq M_4 \subseteq M_5 = S^3V\otimes V$$ where $M_1\cong L(2)$, $M_2/M_1\cong L(0)$, $M_3/M_2\cong L(4)$ and $\{M_4/M_3,M_5/M_4\}\cong \{L(2),L(0)\}$. Which order the two top factors come in is less obvious (I will need to think a bit about it), and whether we actually have $S^3V\otimes V\cong H^0(4)\oplus H^0(2)$ I will also need to think a bit more to figure out.
Added: So, after some further thought, we can actually say a bit more.
First note that as mentioned by Jim Humphreys, we have $S^3V\otimes V\cong L(1)\otimes L(1)\otimes L(1)^{(1)}$ which means that it is self-dual. In particular, we see that our composition series can be chosen to be "symmetric", so we get $M_4/M_3\cong L(0)$ and $M_5/M_4\cong L(2)$ (it is also good to notice that we actually have $M_2\cong H^0(2)$ and $M_5/M_2\cong H^0(4)$ as these are sometimes easier to work with).
We can also show that $S^3V\otimes V$ is indecomposable. In fact, we have $\operatorname{soc}_{SL_2}(S^3V\otimes V) = L(2)$.
To see this, we need a bit more machinery (it might be possible to do this in a more elementary way). Let $G = SL_2$ and let $G_1$ be the first Frobenius kernel of $G$. We let $\lambda = \lambda_0 + p\lambda_1$ be a dominant weight with $\lambda_0 < p$ and use that $L(\lambda) \cong L(\lambda_0)\otimes L(\lambda_1)^{(1)}$. Now we note that $$\Hom_G(L(\lambda),L(1)\otimes L(1)\otimes L(1)^{(1)})$$ $$\cong \Hom_{G/G_1}(L(\lambda_1)^{(1)},\Hom_{G_1}(L(\lambda_0),L(1)\otimes L(1))\otimes L(1)^{(1)})$$ so it is sufficient to show that $\operatorname{soc}_{G_1}(L(1)\otimes L(1)) = L(0)$.
To see this we further note that it will suffice to show that $\operatorname{soc}_G(L(1)\otimes L(1)) = L(0)$ since the $G_1$-socle is a $G$-submodule. But this final part is a simple calculation, as we clearly just need to check that neither $L(1)$ nor $L(2)$ are submodules. That $L(1)$ is not a submodule is clear by parity (all highest weights of composition factors in $L(1)\otimes L(1)$ must be even), and that $L(2)$ is not a submodule is seen by noting that $$\Hom_G(L(2),L(1)\otimes L(1))\cong \Hom_G(L(1),L(1)\otimes L(2))\cong \Hom_G(L(1),L(3))$$ and $L(3)$ is simple (it is the 2'nd Steinberg module as also mentioned by Jim Humphreys).
A few final notes: The above actually shows that as a $G_1$-module, $L(1)\otimes L(1)$ is the injective hull of the trivial module. This is a general fact about $SL_2$ in characteristic $2$, ie, that for all $r$, $St_r\otimes St_r$ is the injective hull of the trivial module as a $G_r$-module (this does not generalize to other groups, nor to other primes).
Also, the conclusion about the module $S^3V\otimes V$ is in fact that it is indecomposable tilting (in the notation from Jantzen, it is denoted $T(4)$).