Can I order my worker to chop and plant forest on the same tile over and over? I would like to do this in order to continuously boost production in the nearest city.
You cannot whatsoever, not even once. When you unlock the Conservation tech you are allowed to plant new forests with builders or Naturalists; however, these newly planted trees cannot be chopped down to boost production in a nearby city.
They will provide the normal +1 bonus for the tile, and you can build lumber mills on top of them, but they are not able to be chopped down for an immediate boost, nor do they provide the same appeal as preexisting forests.