Great thread. Here are my thoughts and responses.
There are two general classes of situation where phosphate removers (PR's from this point forward) could potentially be beneficial -
Recurrent Algae Blooms
There are reports of pool owners who, try as they might, continually fight with algae. They perform a SLAM correctly, pass all three exit criteria and still get algae coming to visit throughout the season. For most of us on TFP, we have to maybe SLAM our pools once and then have gone years without ever seeing algae (I believe chiefwej is the winner of the SLAM longevity contest at 10 years without algae). But there are people for whom the battle with algae is a constant one. In some of these cases, you have to look at extenuating circumstances. It's also helpful to remember that the FC/CYA ratio is NOT a rigorously derived value that is absolute in nature, it's more akin to a statistical value based on a wide sampling of pools. While the 7.5% rule seems to work for most pools, not all pools are covered by it. In those cases where 7.5% does not work to keep algae under control, TFP often recommends going to a higher ratio to keep algae under control (for example, 10%). The problem is, a higher ratio necessarily means a greater chlorine loss rate and more chlorine consumption. In pools with water that is very reactive to algae, determining what role phosphates play can be beneficial. If those types of pools also happen to have high phosphate levels, lowering phosphates, and seeing what impact that has on these "reactive" pools, is a quick and easy experiment to do. If it turns out that lowering the phosphate levels to below 100ppb helps, then that is a net benefit to the pool owner because not only do they get a cleaner pool but also a pool that uses less chlorine. In effect, they save money.
For some pool owners, living with metals in their pool and fill water is a dreadful experience (just ask swampwoman). Some people have to use metal sequestrants continually throughout the season to keep metals in control and the most common type of sequestrant are ones based on phosphonates (HEDP is one example but there are other phosphonates as well). Phosphonates do a good job of sequestering metals but they are susceptible to chlorine oxidation and will eventually breakdown into phosphates. People who fill from well water that is laced with metals (iron being the most common) also try not to fill their pools too much and so keep them covered and minimize water loss. This has the unwanted effect of building up phosphates over time due to the breakdown of sequestrants as phosphates are like calcium and salt, they only leave the water through fresh water exchange. In this case, PR's would be beneficial at helping to reduce phosphate levels for both algae nutrient control and calcium phosphate scaling (although calcium phosphate scaling only occurs at extremely high levels of phosphates).
As for the chemistry, PR's are nothing more than acidic solutions of lanthanum chloride. When you add the LaCl3 to pool water, it almost immediately reacts with the excess carbonate to form the precipitate lanthanum carbonate (that's where the cloudiness comes from). The lanthanum carbonate then reacts more slowly over time with phosphates to form lanthanum phosphate which is also a precipitate. These precipitates get trapped through filtration and removed via a backwash. So, for the most part, all of the lanthanum is removed eventually by backwashing. If you have a sand filter, then you will very likely need to use a clarifier as well because sand filters are not able to remove fine particulates well (although one could try adding DE first to see if that works). If you own a DE filter, you generally don't need to clarifier as it will only make a mess of your DE filter. Cartridge filters may have more of an issue since they're not as fine as a DE filter but you can't easily use clarifiers or backwash them either. I would say with a cartridge filter, you might need the clarifier but it would be good to strip and clean the filter immediately so that it doesn't get gummed up by the clarifier.
As for costs, most good commercial grade PR's run about $100/gal. An initial dose in most pools might be somewhere between 6-12oz. Depending on the source of phosphates, most people would find that a gallon could easily last a few seasons. Only in those situations where the source of phosphates into the pool water is continuos (sequestrant use or municipal water that uses high levels of orthophosphate) would you see a much higher rate of PR use. So I think for the casual user of PR's, the cost is not very high.