The answer now lives on Social feeds troubleshooting, under “Captions with emoji are missing or the feed does not load”.
The database must use the utf8mb4 character set, because utf8 cannot store the emoji that social captions contain. Ask your host to convert the tables (wp_posts, wp_postmeta, wp_options) to utf8mb4 with the utf8mb4_unicode_ci collation, and back up first.
MySQL reports it like this:
Incorrect string value: '...' for column 'option_value'...