I'm currently working on a 2 millions LOC program which never paid its dammed debts, and I weep every day before this holly Big Ball of Mud.
That is, after setting up a test/dummy system, you remove the code, run your tests and try to break it. Perhaps sticking in breakpoints or print lines or whatever..
I doubt you really mean to experiment on the business-critical live system.
Thats unlikely to be very effective anyway. The most obscure code is often a bug fix for some really obscure bugs. Maybe it fixes some weird bug in Swedish XP SP1, or something which you don't find just by removing it and testing. Situation depending, you will need to actually read and understand the code and investigate properly.
Good test coverage help a lot here though - remove something and you'll get a set of failing tests to investigate.
The biggest problem with refactoring and removing code is to explain it to non-techy project managers. If you say that the code or the design is bad, you are saying the guys behind it are bad (you or the previous team), which is not very tactful. I usually try to explain that a software system is a living thing, that need some regular washing up. Also, it is possible to explain that the next features or optimizations will take less time to implement after refactoring.
Works surprisingly often.
Never is a long time.
It's fairly easy for low-skill developers to write code that's time-consuming for high-skill developers to understand. In fact, making simple things hard to understand is pretty much the definition of bad code.
So, you use your judgement. If it's a particularly subtle and important part of the code, spend the extra time to make sure you're not missing anything. If it's not, then just rip it out and don't waste time in a maze of strange control flow, redundant code, useless invariants, and confusing assumptions.