I wouldn't unless its the fizzy variety; Try chilling Pinot Noir instead
Generally I wouldnt. The flavour of wine changes with the temperature - make it too cold and it will loose all/most of its flavour. However, it also depends on what your normal room temperature is. If for example, it is the middle of summer and it is 35C inside then I would recommend putting any red wine in the fridge for 30 min or so, just to reduce the temperature of the wine a little.