In the case of large sets, there is a result of Le Anh Vinh, which states that:

$$I(P,L) \leq q^{-1} |P| |L| + q^{1/2} |P|^{1/2} |L|^{1/2}. $$

Here, $q$ is the order of the finite field. Note, however, that this is worse than trivial when $|P|\times|L|$ is smaller than $q$.

I believe one can deduce an improvement of the form $$I(P,L) \leq |P| |L|^{1/2-\epsilon}$$ from the symmetric case $|P|=|L|$ by using the following argument of Pudlak (see Corollary 2.5, there): Assume that $|P| > |L|$ and select a random subset $P'$ of $P$ of size $L$. The expected number of incidences will be $I(P,L) \frac{|L|}{|P|}$. Let $P'$ denote a set of points at or above the expectation. Now applying the symmetric result to the symmetric incidence problem with $P'$ and $L$, should give $$I(P,L) \frac{|L|}{|P|} \leq I(P',L) \leq |L|^{3/2-\epsilon'}. $$

In the case of large sets, there is also a result of Le Anh Vinh, which states that:

$$I(P,L) \leq q^{-1} |P| |L| + q^{1/2} |P|^{1/2} |L|^{1/2}. $$

Here, $q$ is the order of the finite field. Note, however, that this is worse than trivial when $|P|\times|L|$ is smaller than $q$.