Mowing a wet lawn is never a recommended thing to do. The mower can damage the green blades as it twists and turns on the dampened ground and mower blades can tear instead of cut the grass if they aren't sharp. It is always best to wait until the lawn is dry before cutting. However, sometimes cutting wet grass is a "have to," such as when company is coming, or it's going to rain for days with no hope of a break. If you have to cut during this time, there are a few things you can do to keep your lawn from being damaged.
Set up the bag to capture wet grass as it is mowed. You never want to let it sit on the lawn. It is heavy and clumps together, allowing dead spots to form under it. Bagging might cause clogging, however, so if you don't want to deal with that or don't have a bagger, you will need to rake or blow the grass off the lawn as soon as you are finished mowing.
Raise the mower blade up higher. Cutting less grass means less cuttings to clump on the lawn. This will make removal easier, but still give a nice even look to the lawn.
Check blades to insure they are sharp. You will need blades sharp to cut limp heavy grass.
Set blades at a slower speed to lessen the risk of problems from wet grass, such as clumping inside the deck and clogging.