How to make a film in black and white and add color to some objects?
As you suggest, the scenes are shot in color. There isn't much of a special task for the director, it's almost completely a post-production challenge. In some cases a high contrast take can be made to assist in masking. This would be unusual for scenes involving actors, since the high-con pass wouldn't be well registered. It may also be possible in some scenes to shoot green screen (chroma key) to aid in separating some objects or actors.

The color portions are selectively revealed through masking, and possibly using some green screen and high-con elements. Using a combination of a full color layer and a mono layer derived from it, an artist paints masks that hide or reveal the appropriate layer(s) on a per-object basis. Computerized tracking and tracing can be a great help in this, but in the end it's up to the artist.