Cuts no ice means to have no influence or effect, doesn’t get the job done or impress anyone.
The idiom was first recorded in the US in the 19th century. It may have been a figurative expression right from the start, based on the use of ice in many ways and playing on its hardness and coldness as a metaphor for unresponsiveness or lack of empathy. As to the source, there are several suggestions.
For example, if ice skates aren’t sharp, they don’t allow the wearer to slide easily over the ice. Blunt blades make no impression, therefore they cut no ice.
Another suggestion is that something which has no effect or makes no impression is like a knife too blunt to shave ice off a block. Or it may refer to cutting blocks of ice from a pond or river for use in keeping food cold, so that something or somebody that cuts no ice is useless.
During the 19th and early 20th centuries, crews used large, rough-tooth handsaws to cut big blocks of ice from frozen ponds. It was hard, cold work, so naturally workers would take frequent breaks beside a campfire to warm themselves. Their bosses, eager for the men to get back to work, would remind them — in the most genteel terms, no doubt — that those breaks ‘cut no ice.’
Ice comes into modern life, too. We speak of breaking the ice at parties or other social events, meaning to break down barriers of reserve and get people to enjoy themselves. And let’s not forget that very useful but slippery little object, the ice cube!