The original is ordered in a slightly odd way; usually, the last item in the list is the most important, but I think most people would consider being cheated on of considerably more importance than merely being offended — which could reasonably be a side effect of either of the two previous, well, "offenses". A more natural ordering would be "You offended me, and you lied to me, and you cheated on me." I'll be assuming this for the rest of the answer.
The repetition of "and you" (as well as "me"), as others have noted, makes this list more emphatic. Removing any of those words makes it less so, although not enormously less so; there's still some verbal punch to "You offended, lied to, and cheated on me" (the shortest form that preserves the original meaning). If you remove "me" from the repetition, you cannot also remove the prepositions here, one of which forms a phrasal verb ("cheat on") and the other of which distinguishes between two different meanings of the verb "lie" (deceive vs. rest). That is:
ok You offended, and you lied to, and you cheated on me.
* You offended, and you lied, and you cheated me.
Normally, context would be enough to tell that "You lied." (without "to me") is referring to deception, and indeed it is enough here too to generally grasp that meaning. Grammatically, though, if you omit all but one of the objects in a repetitive sentence like this, they're all tied to the same object more closely, and as Hellion has noted in comments elsewhere, "lied me" doesn't work here. The meaning can be reconstructed despite the poor structure, but it's best to avoid that mistake in the first place.
It's also often best to avoid omitting repeated objects (or subjects), even though it's grammatical, because it requires more thought to understand properly. It's also more formal, so in this context, it sounds odd to be using such a precise economy of words in what is certainly an emotional statement.
You can omit commas in the original without any real problem (perhaps a slight loss of dramatic pauses), but if you take out the conjunctions this no longer works:
ok You offended me and you lied to me and you cheated on me.
* You offended me lied to me you cheated on me.
(If you want to get across the idea of someone incoherent with rage, so upset they simply can't talk grammatically, the latter might work. Otherwise, no.)