From as near as I can figure, it seems like there is basically no difference to being Asian for Law school admissions. If there is, it might be a very small negative effect at top schools, but I'm unsure if even that exists. Reposting what I basically researched last cycle:
I did a little research into this when I was applying as I was curious about this question too. The fact that Asians are ORM for UG admissions is definitely true (there was that research study awhile back that found Asian applicants experienced the equivalent of -160 points on their SAT (out of 1600) when applying to college), but this definitely does not seem to carry over to Law School admissions. Anyways here was my best estimate from just examining several sources and making a guess based on what I know:
Asians applying to law school experience are basically equivalent to White applicants, although perhaps there is a slightly negative effect at the highest tier schools (HYS).
Again, this is highly speculative as there are tons of confounding factors, but the basic idea is that ABA data shows a trend of the top 3 law schools perhaps being a little more selective for Asian applicants.
The Data (ABA latest statistics for law school population):
Harvard: 10.1% Asian, 11.3% African-American
Yale: 11.6% Asian, 7.5% AA
Stanford: 12.6% Asian, 10.8% AA
Columbia University: 15.1% Asian, 8.1% AA
Chicago: 11.5% Asian, 6.3% AA
NYU: 10.5% Asian, 6.5% AA
So, for example, Harvard UG has much more Asian students then HLS (around double the percentage). The African-American numbers are just for comparison as they show that, overall, the top 3 law school have a greater share (percentage-wise) of minority applicants. Thus, for example, it seems like Harvard is the only school where there are more African-American's then Asian's attending. This is despite the fact that Harvard UG is approx. 9% AA and approx. 18% Asian, roughly speaking.
Now for the stuff that could be affecting this numbers enough to mean that there is no negative association with Asian applicants at all:
-Asians are more likely to go towards science/ medical track fields than law school. This has been shown in quite a lot of research and definitely contributes to fewer Asian law school applicants. The caveat is that there should still be quite a lot of top Asian law school applicants considering the percentage of Asians in top UG's.
-Location, school preference, etc. : For example, Stanford has more Asian applicants because of its location. Columbia blows NYU out of the water, 15% to 10%, perhaps because of greater emphasize on the Columbia Ivy League name over NYU. However, this raises the fact that Harvard might experience the same effect of increased applicants. (Also might be true for other minorities, remember seeing something about AA applicants preferring H out of HYS too).
-The numbers are spread out that way due to school preferences. For example, Chicago is known for not being very URM friendly, which is probably affecting its numbers.
Anyways take of it what you will, the actual data I can find isn't very conclusive or glaring... in the end I think there is little to no effect so you probably don't have to worry too much about it. I don't think I saw an effect in my admissions cycle either.