How do you remove dry bits (from old rubber seals) that have "caked" onto chrome pieces without scratching the surface?

If the rubber is disintegrating such that it is coming apart you will likely need to replace the seal/rubber. To get the crud off the chrome I would try some type of mild solvent cleaner first. This sounds as if it is slightly beyond the call of duty for Meguiar's Rubber and Vinyl cleaner but you can always give the basics a try. Ditto for Vinylex. If those first few don't work, gradually keep trying more severe solvents on the rubber until you find something that does work. Do be careful with something like Simple Green though. Goop may also be helpful.

Whatever product(s) you use, be sure to apply/remove with only a 100% cotton terry. Keep "turning" the terry to be using a clean surface so that you don't scratch anything.

Hope this helps...Lee