In the first pic, I'm seeing elm, maples, oxalis, poke sallet... nothing especially dangerous, as long as you avoid eating them... In the second pic I see grape vines, oxalis, a sweet gum, and possibly richardia under the board... again, nothing especially hazardous... I'd leave them be.
There are three main plants that can cause a bad skin rash just from touching the plant. Poison ivy, poison oak, and poison sumac are the three main plants that can cause anything from a mild to severe itchy rash in most people. The plants in the picture are not any of those three plants.
Now, that doesn't mean these are completely "safe" plants. There are MANY, MANY plants that can be poisonous if a young child actually EATS the plant. For that reason, it is usually best to pull weeds like these and throw them away. You can also use a weed killing spray to keep the weeds from growing in your rocks.
Basically, I am very confident in telling you that you can touch these plants to dispose of them without worry.