Cash runs don't have to be completed to finish the game in any%. They are just used to teleport around the map to get to a location quicker.

For 100%, you only have to complete one cash run challenge under the gold time.

You won't need to delete the cash runs from a previous playthrough if you don't use them. I don't see why you won't use them in an any% run because they will save at least several minutes of travel time in the run.