"Would you like to try our red wine?"
With wines it is acceptable, at least nowadays. 试 has the meaning of Test and wine-tasting is generally translated as 试酒. Consequently, it is quite common now to use 试, but in the meaning of To Taste as in French déguster, not as in goûter.
There is no particular reason why it would not be acceptable. Certainly it would depend on the situation and asking in such a plain question of the counterparty's desire may not always be appropriate (as it would with any other language). e.g. with wrong intonation and expression it may want to say What? You want to drink our wine? How dare you!
I find it a bit different. "Would you like to drink our red wine?" is different from ""Would you like to try our red wine?" After trying their wine, I may not want to drink it! :-)