Define "plate" from Sweeney Todd musical

Solution 1:

Here, the meaning is, "The [vulture of the law] removed the barber from [the vulture's] plate."

This is a formulation of the idiom off one's plate:

No longer a matter of one's responsibility and concern

(Compare with the related too much on one's plate.)

In this idiomatic context, one's "plate" is the set of concerns one is trying to resolve.

The judge (the "vulture of the law") wanted to get access to the barber's wife; the barber himself was a matter of concern for the judge in achieving this end. When the judge eliminated the barber as a concern, he "removed the barber from his plate".

Solution 2:

Considering that it's a musical involving human meat pies, I think it's a metaphor insinuating that, to the "vulture of the law", people were just food on a platter. By removing Sweeney from his "plate," the girl was then left alone and desperate.