Would you eliminate a candidate from the interview process if they had little or no knowledge of your company?

In a word yep 

If you can't be bothered to do even some basic fundamental research about a prospective employer, then you are displaying either laziness or a complete lack of initiative.

Two qualities I would definitely steer clear of 

If I were to go for another interview I would certainly want to know what I was letting myself in for as well.. Although I was lucky enough to already know just about everything there was to know about my present job as I did warranty jobs with them all the time so knew everyone pretty well

I think it is courteous to learn something about the company but what irritates me are the companies with "mantras" that are often displayed on the reception area wall.   I once attended an interview with a well-known supermarket company for an admin role so had a look at their website to understand the company values, which can be different from what you actually witness as a customer.

Wasn't asked anything about company values but had to demonstrate how I would deal with different situations and if there was anything in the past where I learned from a mistake, either committed or averted.  They assumed, rightly or wrongly, that I knew their company but they had also taken the time to read my CV and ask sensible questions.  Unlike an interview many years ago, where I was asked some ridiculous questions such as if I died tomorrow, what would I like inscribed on my headstone.  I resisted the temptation to reply "I don't care as I won't be able to see it".