In general we can consider a differential $d_t$ which depends polynomially on $t$ and whose value at $t = 0$ is our original differential. Simple examples show that we can pick up extra cohomology at special values of $t$. For example we can take a complex of the form $C^0 \xrightarrow{d_t} C^1$ with $d_t = \left[ \begin{array}{cc} 1 & 0 \\ 0 & t \end{array} \right]$. The dimension of $H^0(C) = \text{ker}(d_t)$ jumps from $0$ to $1$ at $t = 0$ and so does the dimension of $H^1(C) = \text{coker}(d_t)$.

However, I claim that at least for complexes of finite-dimensional vector spaces we can never lose cohomology at special values of $t$ (I'll be more specific about what I mean by this). In particular, if the complex is acyclic at special values of $t$ then it is acyclic everywhere. The example I have in mind, which I learned about very recently, is a particular complex associated to the family of algebras

$$U_t(\mathfrak{g}) = T(\mathfrak{g})/(X \otimes Y - Y \otimes X - t [X, Y])$$

where $\mathfrak{g}$ is a Lie algebra. $U_t(\mathfrak{g})$ is isomorphic to the universal enveloping algebra for $t \neq 0$ but isomorphic to the symmetric algebra for $t = 0$, and the corresponding complex is isomorphic to the Chevalley-Eilenberg complex of $\mathfrak{g}$ for $t \neq 0$ but isomorphic to the Koszul complex of $S(\mathfrak{g})$ for $t = 0$. The claim above implies that the acyclicity of the latter implies the acyclicity of the former. (In order to think of my complex as the same complex with a varying differential as $t$ varies I need PBW.)

So, here's what I mean by special: the action of the differential $d_t$ in a particular degree $C^n \to C^{n+1}$ induces a map $\Lambda^k(C^n) \to \Lambda^k(C^{n+1})$ for all $k \ge 0$. The dimension $\dim \text{im}(d_t)$ is the largest $k$ such that this map is nonzero. The condition that this map is nonzero, for fixed $k$, is a Zariski open condition in $t$ since it is defined by the nonvanishing of at least one $k \times k$ minor of $d_t$. Hence $\dim \text{im}(d_t)$ generically takes on a single value, and at a possibly empty exceptional set of points defined by the vanishing of the $k \times k$ minors of $d_t$ for $k = \dim \text{im}(d_t)$ it takes on a smaller value. And of course $\dim \text{ker}(d_t) + \dim \text{im}(d_t) = \dim C^n$ is fixed, so $\dim \text{ker}(d_t)$ generically takes on a single value and takes on a larger value on a possibly empty exceptional set.

In the above example the behavior for $t \neq 0$ is uniform so the only possible special point is $t = 0$. In general it might be harder to say what the special points are.