Logjam on Elliptic Curves?
Let's recall how discrete logarithms are solved in strong elliptic curve groups. The basic idea is to iteratively walk through many combinations of the form $x_i = a_iP + b_iQ$ until we find a distinguished one, i.e., one that shares some common property (like the lowest $k$ bits of $x_i$ set to 0). We accumulate enough distinguished points until we find a collision.

When we find two distinct pairs $(a_1, b_1)$, $(a_2, b_2)$ that evaluate to the same value $x_1 = x_2$, we recover the logarithm by finding $Q = dP = \frac{a_2 - a_1}{b_1 - b_2}P$, which follows from $a_1P + b_1Q$ $=$ $a_2P + b_2Q$. We expect this to happen after $\sqrt{\pi n / 2}$ elliptic curve group operations.

Since by the time we get to the second discrete log we already know the first one, we can reuse the distinguished points found during that log computation. This slightly increases the chances of finding a collision. Suppose we have a hit $a_1P + b_1Q_1$ $=$ $a_2P + b_2Q_2$, where $Q_1$ was the first discrete logarithm target, and $Q_2$ is the current one. Rewrite this as $(a_1 + b_1d_1)P$ $=$ $a_2P + b_2Q_2$, where $d_1$ is the first discrete logarithm. Now we have $d_2 = \frac{a_1 - a_2 + b_1d_1}{b_2}$.

This accelerates the second logarithm by a factor of $2$, the third one by a factor of $8/3$, and the $i$th one by a factor $\approx \sqrt{i}$ compared to if they were solved independently. So as you see, the benefits of precomputation exist, but are much more limited than in the number field sieve case.