In order to actually kill ich with temperature, you would have to raise the temp to almost 90F. Increasing the temperature just speeds up the life cycle of the ich so you can treat it faster. In a cold-water tank, the life cycle can take several weeks, but at temperatures around 84-86F it will complete in just a few days (four to five at 85F).
Ich is a protozoa, and its life cycle has distinct stages. Only during one of the stages is it actually vulnerable to any treatment - the free-swimming stage.
Ich on the fish (called a trophont) is encased in the fish's own epithelium. It is not treatable. Once it completes its feeding, it is shed and swims freely about the tank for a few hours - during this time it is vulnerable. Then it settles in the substrate and encases itself and begins to divide (at this point it's called a tomont, or a cyst, and it is not vulnerable). Each cysts produces hundreds of theronts which escape when the cyst opens and swim freely about the tank until they find a host fish and attach. They must find a host within 48 hours or they will die. This stage is when they are most suceptible to treatment. No matter what you are treating with - salt or meds, they are immune to treatment when they are on the fish and when they are in the substrate (and filter).
Ich do not burn the fish's gills. Ich like to attach in the gills because they are "soft", have the best blood supply, and don't have thick mucus coating. You can't see ich in the gills (or newly attached trophonts), so there is NO way of looking at a fish and determining if it has ich or not. Some fish are carriers and avoid large-scale infections but have chronic gill ich. Also, the only time ich can feed is when attached to a fish in the trophont stage. When ich are free swimming they must find a host to feed in about 48 hours (at normal aquaria temperature) or they will die. Also, when ich is in the tomont (cyst) stage, they have a limited amount of stored energy which is all used for reproduction. They can not live in the tank indefinitely unless they are on a host. Sudden outbreaks that are attributed to "dormant" ich in the gravel or the filter are most likely flare-ups of a chronic gill infestation when stressful conditions arise.
Since the ich are obligated to attach to a host, moving the fish around between several tanks is another option to kill ich. It's labor-intensive and requires multiple tanks, but it is 100% effective in eradicating ich. However, it may produce additional stress for the fish because it involves netting and moving them multiple times.
Gravel vac and water changes can greatly reduce the number of tomonts and free-swimming ich (theronts). Be sure to let your equipment dry completely between uses because that will allow any ich present to dry out and it will die.