I do, because my tapwater is loaded with sodium bicarbonate and has essentially no calcium and magnesium. I have no rain gutter, and I have a couple of plastic trash cans located at a corner where the runoff is concentrated. If it is just a light rain, and, if it hasn't rained in a while, the water that comes off is dirty and somewhat foamy. If there was something really bad in that water, I suppose it would kill the grass, and I have never seen that. Just the same, I don't use that water. When there is a larger amount of rain, I throw out the initial, dirty water and keep the cleaner water that comes off later. I have had no problems with it.
In the summer you don't want to leave the water standing in the trash cans and breed mosquitoes. Dump out what you do not immediately collect.