Currently my employer in Illinois is trying to figure out a way to eliminate a horrible manager and promote a supervisor position. This manager would either be terminated, due to bad management and time falsification (which managers do not have time cards, they write down their hours and send via email), or be demoted. What can they do to prove this manager is bad and needs the demotion/termination?

You allow your employer to handle this situation. It is none of your business.

Demotion never works. Firing is the best option.
Your employer needs written policy so that he or she doesn't face discrimination lawsuits. Such policy can't break federal or state law, but many situations aren't addressed by law, and even law gets muddled easily when an employee decides to charge that racism, agism, or sexism are the real reason. Even something like being there too long and making too much money, one very common reason for getting rid of someone in favor of a new one willing to work for much less.

An employer doesn't, in theory, have to prove anything when their reasons are bad management. Just make a list of poor work results and the word of underlings.
Obviously falsifying hours worked can be an easy ground for immediate dismissal. How to prove it in a law court? Time stamped cameras. Testimony of other workers, and the lack of solid evidence of work during the time they weren't there. Social sites like Facebook can sometimes reveal a lot.

I wouldn't make a big deal about all the 'evidence' unless this is a union worker, or someone you suspect will sue. But this is why most policy includes specific steps to being fired, usually a 90 day probation period when first hired, and X number of warnings of X duration, in person and in writing, in the file, before dismissal.

Your employer needs a real HR employee who knows all this and how to do it.

As was mentioned. Managers are generally never Union... and except for Executives... rarely if ever have contracts. Meaning all they have to say is they weren't performing up to expectations. And that's it... they are gone.

You really need to stay out of this because it's the responsibility of their superiors and the HR dept. And you will bring negative attention to yourself as well.

Assuming you aren't speaking of yourself.

That's why bosses get paid the big bucks to make such decisions. Are you trying to help the boss to get this fellow's job or what?