Indian mock strawberry, Potentilla indica, is a non-native weed only slightly related to “real” strawberry. It’s flowers are yellow while those of edible strawberry are white or pink. The fruit of mock strawberry is not poisonous but the taste is bland and dry. You won’t really like to eat them.
Plants are easy to pull, so that should be your first defense.
A spot spray of organic non-selective weed killer (click for information) or synthetic non-selective weed killer (click for sources) would kill a plant.
For your lawn, one of the broad spectrum herbicides like Bonide Weed Beater (click for sources), Ortho MAX (click for sources) or Bayer Season Long Control (click for sources) could give excellent control.