Analyze the Condition of the Ceiling
Inspect the ceiling to determine what you need to do to repair and repaint it. Assess whether the ceiling and popcorn texture is intact but just stained where radiant heat lines run above it. Determine if there is actual water or condensation damage from the lines. Note whether the popcorn is intact or failing along seams in the ceiling or coming off in chunks in multiple places.
Find out what kind of materials were used in your popcorn ceiling. Some contain asbestos. Removing this kind of ceiling will require asbestos abatement equipment, procedures and permits. Most modern popcorn textures are made from polystyrene pellets but some use perlite as the texturing compound. Before it was banned for use in construction materials, asbestos was a common component of popcorn ceilings so send a sample of yours to a lab if you suspect it might contain asbestos.
Repair the radiant heating system above the ceiling if necessary before you proceed to eliminate leaks or excess condensation that are the source of ceiling problems.
Possible Repair and Refinishing Approaches for Modest Damage
Lay down tarps and cover everything exposed.
Apply a high quality stain blocking primer and two coats of quality paint after all issues from the heating system are resolved and you have determined the coating does not contain asbestos. Because most popcorn ceilings were applied without a proper sealer and primer, they tend to come off in pieces on roller covers. Using a thick foam roller, rather than a fabric one, may help reduce surface tension. Spraying is a better option but use the lowest pressure the equipment and paint will allow.
Prime the newly exposed surfaces to be patched.
Apply replacement popcorn finish to bare spots. You can buy aerosol cans of repair material that are supposed to cover six square feet each. The cans are expensive ($12 to $16) and explode a 15-second burst of texture and adhesive all over the place. A better option is to buy the polystyrene pellets, mix them with a high bond primer, and blend in the patches as best you can with a large 4-inch brush.
Rent the texture equipment, or hire a professional, to lay another coat of texture over the existing ceiling surface if you are certain what is left is adhering nicely to the ceiling. Adding the weight of another layer if there is no adhesion with the base layer will just postpone major problems.
Scrape all the texture off the ceiling if it is failing in multiple locations. Use a wide drywall blade and you will be surprised at how fast it goes.
Sand, prime and repaint.