"I also do not drink tea."
I asked someone who is not a native but fluent speaker of Mandarin and this is his answer. "It's actually not acceptable to reverse the order--it should be 我也不喝茶 with 也 before 不. For what my limited intuition is worth, I think it's possible to have some scenario with the reversed order with really restricted use. Maybe something like 我不是也喝茶, "I am not also drinking tea".
This is probably due to co-articulation (the influence of preceding or following segments in the sound stream). For example, in English the g in big sounds more like a k before kite (big kite) than before goat (big goat). Its just because our articulators (tongue etc) are anticipating the next sound to produce (or remembering one just pronounced).