I agree with Tdol that if it isn't true, it's not a fact. However, if you had told someone something earlier in the day which you had claimed was a fact and you were now going back to them to tell them that you had discovered that you were wrong, you might say "Sorry, the "fact" ... is not true" or "Sorry, that "fact" has turned out not to be a fact!"
My personal opinion is that you need neither "that" nor "which".
But The fact I told you isn't true doesn't work for me. You could say What I said was a fact wasn't true, but without saying something like this, the sentence is a mess to me because it contradicts itself if it uses the word fact.