Which perl version are you using, which OS?
I don't see a specific problem with the code. It should have a use strict; at the beginning, though.
How long is a typical JSON string?
Can you create a sample input that will show memory increase? For example, instead of reading in the file, just loop over the same line a couple of 1000 times and watch memory consumption.
The problem the others are mentioning doesn't look like a problem to me since $jp falls out of scope after every loop iteration.
Memory stays the same for me if I run this 500,000 times on an example line.

edit: specifically, how many array elements does a typical structure have? my assumption was that you have many lines, but maybe your lines/JSON strings are very big?