Removing an anchor from a concrete wall is not like removing hollow wall anchors. Sleeve and wedge anchors expand near the base of the anchor. The bolt in these anchors will not remove cleanly from the wall. The alternative is to set the anchor deep enough into the wall that a patch can cover the hole. Another type of concrete anchor is the Tapcon brand screw anchor. The screw anchor cuts threads into the concrete as the anchor is set. The Tapcon is the simplest to remove because it threads out just like any other screw.
Remove the nut around the wedge anchor bolt with a socket wrench. Pull off the washer between the nut and the concrete wall.
Cut the wedge anchor bolt flush with the wall with either a hacksaw or Dremel tool.
Place the tip of an awl on the centre of the wedge anchor bolt and tap the end of the awl with a hammer until the end of the anchor is recessed into the wall. This will give you room to patch over the wall.
Grasp the head of the Tapcon screw with the appropriate screwdriver.
Turn the screw counterclockwise just like a regular screw to remove it from the wall.
Grasp the end of the screw periodically with a pair of pliers and pull out if the screw begins to turn without moving away from the wall. The pulling will add pressure to the screw. Alternate between pulling and turning to remove the screw.
Remove the lag shield bolt with a socket wrench.
Insert a large Phillips-head screwdriver into the lag shield where the bolt would insert. Break the lag shield into two pieces. Use a screwdriver or other straight steel object that is slightly larger than the bottom of the lag shield.
Tap the screwdriver with a hammer to force the lag shield to separate.
Pull each piece of the lag shield out of the wall with needlenose pliers.