Some chrome trim have a clear protective wrapper that peels off after grouting. I am assuming yours doesn't. Make sure you use a fine grout. I personally use Ardex FS grout (comes in about 32 different colours available from CTD (aka
Tile Depot). However, test your grout on a small piece of off cut trim before you commence grouting. Make sure it's wall grout (as this contains much finer sand). Floor grout contains thicker sand and will deffo scratch your chrome.
Don't let the grout sit on the trim any longer than it has to. i.e. wipe off any excess with a damp microfibre cloth and buff up with a dry one.
Realistically, the chances are you will not damage your trim as long as you don't wipe acid on it!