I've been gardening a long time and was sure that the answer to that question was in the negative. But once again it turns out that no matter how many years you've been growing and mowing, you always have things to learn.
Obviously, there are good reasons not to mow the lawn when it's wet: Damp grass is slippery to walk on, and clippings tend to clump, which doesn't look good, and, depending on how thick they are, may mean you have to rake them up. That's no fun for you and not so good for the lawn, either, since those clippings provide free fertilizer.
Perdue University takes the long view -- when it rains a lot, you're going to have to mow during periods when the rain has stopped, whether the grass has dried or not. So they offer six pointers for mowing wet grass.
As this discussion on Garden Web points out, if you live in the Pacific Northwest, you might never mow the lawn if you waited for it to dry out. Oiling the underside the mower's deck or spraying it with a silicone spray may help.
But does mowing the lawn when it's soaked cause harm to the grass? Kansas State says no.: "Contrary to popular belief, mowing wet grass is better than letting grass grow too tall – if the mower blades are sharp." But KS experts do point out one more disadvantage of mowing wet grass: The chlorophyll in wet grass will stain everything it touches -- you, the driveway, the house if the clippings are blown on it, etc.
And, as "briwei" reports, if the ground is soggy enough that you sink in as you walk, it's not going to work well.
Naturally, it wouldn't be safe to use an electric mower with extension cord when the grass is damp. But you already knew that.
So there you have it. There are good reasons not mowing when the lawn is soaked, but it doesn't necessarily harm the grass and it can be done if you need to and follow some precautions. Nice to know, I think.