Hi, as a senior paralegal at my firm, I get to hear the inside scoop on a lot of the resume review processes. At our particular firm, we have attorneys from the two major law schools in San Diego, so both of them know the score as far as making moot court teams.
At least at our firm, putting how far you got in a competition doesn't hurt you. It is one way to set yourself apart from the rest of the resumes. How much it helps you depends on how far you went, the further you go, the more it helps. "participant" or "top 32" tends to draw a few chuckles and comments about how hard everyone tries to impress the hiring partners. Once you get on the "team" it can lead to being higher up on the call back list.
We're not BigLaw by any stretch, so at some of the stuffier firms, the situation is likely a bit different.