A hedge trimmer cannot tell the difference between a wet hedge and a dry hedge: it can cut through either with little difficulty. With that said, some safety and maintenance issues need to be addressed before you go out on a wet day to trim your hedges. Addressing these problems will ensure you don't damage your hedge trimmer or injure yourself. These issues apply only to gas-powered hedge trimmers; electric trimmers should never be used in wet conditions.
Ensure Cutting Teeth are Sharp
You'll need to make sure the cutting teeth on the top and underside of every tooth is sharp. Cutting when the hedge is wet will reduce the traction of the teeth on the leaves and branches. Dull teeth will also reduce the traction, so you need to take the trimmer in to get sharpened or sharpen it with a crosscut flat file at home. You shouldn't have any bent or damage teeth because you need as much traction as possible to slice through the branches.
You also need to ensure the teeth are properly lubricated, as water can decrease the mobility of the cutting teeth's motion. This effect is minor, but combined with dull cutting teeth it can pose a more serious problem to the gears inside the cutting head. Also, the proper amount of lubrication will prevent water from entering more sensitive parts like the drive area and power head. Lubricate the teeth with hedge-trimmer teeth lubricant as specified in your operator's manual. You can use a rag to apply the grease to the teeth so you don't cut yourself.
Protecting Against Rust
After you've finished your trimming you'll want to ensure no problems later on. Water, if left on the metal, can oxidize and start a rust buildup, which will ruin your cutting teeth and other parts later on. To prevent rust you will need to dry any metal areas with a rag once you've finished or leave it out in the sun to dry for several hours. Don't put the hedge trimmer into a cold, dark storage area while it's still wet.
The biggest issue when cutting while it's wet is that the ground is also likely to be wet. This can cause a dangerous situation, as the cutter's footing will be greatly reduced. Unlike chainsaws, many hedge trimmers don't have an emergency brake that will stop the cutting teeth if you fall. Falling onto a hedge trimmer can serious injury and even death. Take all safety precautions -- wear boots with good traction, never cut alone, and always ensure you have good footing before pressing the throttle and making a cut.