This is really a comment on Donu Arapura's answer, but it seemed large enough to deserve it's own post. Working again in the case of $GL_1$, Simpson considers three spaces:
$M_{betti}$: The space of $\mathbb{C}^*$-local systems on $X$. If you like, you can think of this as smooth complex line bundles with an integrable connection.
$M_{DR}$: The space of holomorphic line bundles $L$ on $X$, equipped with an integrable holomorphic connection. (Being compatible with a holomorphic connection forces $c_1(L)$ to be $0$ in $H^2(X)$.)
$M_{Dol}$: The space of holomorphic line bundles $L$ on $X$, with $c_1(L)= 0$ in $H^2(X)$, and equipped with an $\mathrm{End}(L)$-valued $1$-form. $\mathrm{End}(L)$ is naturally isomorphic to $\mathcal{O}$, so this is just a $1$-form, but it is the $\mathrm{End}(L)$ version which generalizes to higher $GL_n$.
The first space is $\mathrm{Hom}(\pi_1(X), \mathbb{C}^*) = H^1(X, \mathbb{Z}) \otimes \mathbb{C}^*$. In this latter form, it has a natural algebraic structure, as a multiplicative algebraic group.
The second space is an affine bundle over $\mathrm{Pic}^0(X)$. Each fiber is a torsor for $H^0(X, \Omega^1)$, so we can describe this space by giving a class in $H^1(\mathrm{Pic}^0(X), \mathcal{O}) \otimes H^0(X, \Omega^1)$. By GAGA, this cohomology group on $\mathrm{Pic}$ is the same algebraically or analytically; viewing it algebraically, we get an algebraic structure on $M_{DR}$.
The third space is simply $\mathrm{Pic}^0(X) \times H^0(X, \Omega^1)$ (for larger $n$, this vector bundle can be nontrivial). For obvious reasons, this has an algebraic structure.
The relations between these spaces are the following: All three are diffeomorphic. $M_{betti}$ and $M_{DR}$ are isomorphic as complex analytic varieties, but have different algebraic structure. $M_{Dol}$ and $M_{DR}$ are not isomorphic as complex analytic varieties, rather, $M_{Dol}$ is the vector bundle for which the affine bundle $M_{DR}$ is a torsor.
You might enjoy writing this all down in coordinates for $X$ an elliptic curve. As smooth manifolds, all three spaces should be $(\mathbb{C}^*)^2$.
As already pointed out, the Hodge numbers may go up under reduction modulo $p$. On the other hand, let me also point out that the situation can be controlled:
1.) For all $p$, where $\overline{X}_p$ is smooth, the $\ell$-adic Betti numbers of $X$ and $\overline{X}_p$ are the same.
2.) Now, by the universal coefficient formula relating crystalline and deRham cohomology, we have for all $i$ short exact sequences
$$
0 \to H^i_{cris}(\overline{X}_p/W)/p\to
$$
$$
H^i_{dR}(\overline{X}_p/k_p)\to {\rm Tor}_1^{W(k_p)}(H_{cris}^{i+1}(\overline{X}_p/W),k_p)\to 0
$$
where $k_p={\cal O}_K/p$. Now, if $\overline{X}_p$ has torsion-free crystalline cohomology, then the term on the right is zero, and the term on the left is a $k_p$-vector space of dimension equal to the $i$.th $\ell$-adic Betti number.
Then, the Fr\"olicher spectral sequence relating Hodge- and deRham-cohomology degenerates at $E_2$, we have that $\sum_{p+q=i}h^{p,q}$ is equal to the $i$.th $\ell$-adic Betti number. Thus, simply for dimension reasons, the Hodge numbers of $X$ and $\overline{X}_p$.
The upshot is that torsion in crystalline cohomology of $\overline{X}_p$ detects and controls the differences in Hodge numbers of $X$ and $\overline{X}_p$. For almost all $p\in {\rm Spec} {\cal O}_K$, the reduction $\overline{X}_p$ will be smooth and will have torsion-free crystalline cohomology.
Best Answer
Without flatness you have little chance for this to even get off the ground: Let $f:X\to Y$ be the blow up of a smooth (closed) point of $Y$. Then all fibers except one consist of a single point, while the special fiber is a $\mathbb P^n$. That will have non-zero Hodge numbers that the others can't even dream about.
So, assume $f$ is flat, then if $X$ is smooth, then $\Omega_X$ is still locally free and so it is flat over $Y$ and hence you get that $\dim H^q(X_y, (\Omega^p_X)_y)$ is semi-continuous. This works in any characteristic and it does not have anything to do with Hodge theory. The shortcoming of this is that you're actually not getting $\dim H^q(X_y, \Omega^p_{X_y})$ even in the smooth case, because for that you would need $\Omega_{X/Y}$, but that's not flat and in some sense not the right object to consider.
If $Y$ is also smooth, then $\Omega_Y$ is locally free and if $f$ is dominant, then you still have the short exact sequence $$ 0\to f^*\Omega_Y\to \Omega_X\to \Omega_{X/Y}\to 0 $$ The trouble is that the sheaf on the right is not locally free, so when you take exterior products, then it gets kind of tricky. One possibility is to construct complexes that behave very similarly to $\Omega_{X/Y}^p$ with respect to $\Omega_X^p$ and $\Omega_Y^p$. This is done in this paper. The primary goal of the paper is not what you want and I am not sure that you actually get semi-continuity, but you can at least check out the construction. A general version of that construction is in this paper. (Sorry, neither of them are on arXiv).
Also, for singular fibers $\Omega_{X_y}$ is not the "right" thing to look at. One can look at the objects that come from the Deligne-Du Bois complex and do Hodge theory for singular varieties. Then the restriction becomes a little tricky, because those are objects in a derived category and restriction is not exact, so you need to do something else. Completion along the fiber gives the right thing, but I don't know if there is a semi-continuity theorem using completion instead of restriction. That might be an interesting question to contemplate. This is related to the paper I linked above. See the references in that for more details.