Say I have a cubic Bezier curve, with a starting point s, an ending point e and control points c1 and c2. Given t between 0-1, I want to find the equation of the subsection of the curve between 0 and t.
Is this possible without being too computationally expensive?
Yes, it's possible. The deCasteljau algorithm divides a Bezier curve into two. If you search, you'll find lots of references.
Let me change your notation: let the points of the curve be $A$, $B$, $C$, $D$. So, $A$ is the start point (where $t=0$), $D$ is the end-point (where $t=1$), and $B$ and $C$ are the control points adjacent to $A$ and $D$ respectively.
Suppose you are given a value $t$ at which you want to split the curve. The deCasteljau algorithm tells you to do the following calculations:
$$L = (1-t)A + tB \quad ; \quad M = (1-t)B + tC \quad ; \quad N = (1-t)C + tD $$ $$P = (1-t)L + tM \quad ; \quad Q = (1-t)M + tN $$ $$R = (1-t)P + tQ$$
Then the control points of the "left" portion of the Bezier curve (the piece from $0$ to $t$) are $A$, $L$, $P$, $R$. And, if you're interested, the control points of the "right" portion of the Bezier curve (the piece from $t$ to $1$) are $R$, $Q$, $N$, $D$.
This picture is not accurate, but it might be helpful: