If it's from rain or morning dew, it is never safe to mow wet grass, especially with a push mower. Wet grass can be slippery, which can lead to twisted ankles on rough terrain or steeper incline -- and more serious injuries if you fall onto your push mower. Wet grass clumps up in the discharge chute of your mower, making the engine work harder to cut grass, which can lead to burnout, broken belts, and faster-dulling blades. Clumping will also require you to stop frequently to clear chute clogs, which can lead to nasty cuts -- or lost fingers and hands -- if the blades aren't fully stopped.
Wet grass clumps can also kill portions of your lawn, since they can gather in big piles, cutting off air and sunlight. And you won't be able to get an even cut on wet grass, due to the blades quickly becoming duller in addition to the clumping. You'll get tall stripes and patches like you missed entire passes, which means you'll have to essentially mow your lawn twice.
In the end, it's best to wait for your yard to fully dry before cutting the grass to keep your push mower in good shape and all of your limbs intact.