My kitchen countertop is tile, and my old cast-iron sink has begun to rust. Can the sink be replaced without removing the tile around it? Or is there some product that I can use to sand down the sink and recoat it?
You should be able to get that sink out without taking your countertop apart. First, use a tile saw to grind or saw out grout to create - or increase - the gap between the sink and the tile. Remove plumbing carefully and work that sink loose. There's a chance you'll damage some tile the process, but if you save the pieces you can reglue and regrout that tile.
In terms of whether resurfacing or relining the sink is an option, there's unfortunately not a product that I'd recommend. Cast-iron sink finishes are baked on by the manufacturer, and there's no over-the-counter product that comes lose to replicating that level of durability.
Good luck! Post before and after pics to The Money Pit's Facebook page.