Translation:Do not look at me, I do not drink tea.
It looks like "mat" is for commanding someone not to do something, whereas in this sentence "nahin" is used in a statement rather than a command. "Don't drink tea!" would be a command using "mat" but "I don't drink tea" is a statement using "nahin." If I understand correctly.
(Native speaker here.) Not really, unless current Hindi-speakers in India have adopted this meaning from English. If I heard this in a passing context I would be confused about the first part and not sure why the speaker said 'don't look at me'. Just 'I don't drink tea' is good enough.
It's likely because of 'chai'.
In English (at least in the US), 'chai' usually refers to an Indian spiced tea with cinnamon, cardamom, cloves, ginger etc added.
In Hindi, चाय (which comes from the original Chinese word for tea) is just 'tea' and if you ask for it, you'll be given regular black tea (with milk and sugar). If you want the familiar spiced drink, you have to ask for 'मसाला चाय' (literally, 'spice tea').