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When someone asks for a sip of your drink, be it alcoholic or non-alcoholic, telling them to church wine it is saying just touch the liquid to your lips and that's all you're getting. This comes from the eucharist part of a Catholic mass in which wine is presented to be gently sipped, not drank.
"Can I have a sip of your beer man?"

"Ugh, alright, but church wine that shit because I don't have much left."
by StealsyourstereO January 02, 2006
Mixed alcoholic drink. Consists of Banker's Club Vodka and cran-grape juice. Consumed by the poor, college students, and church goers.
"Hey bro, do you have anything to drink?"
"I'm all out of natty light, but I can make you a church wine."
by TheRedBaron1916 February 04, 2010