New emoji’s / emoticons are unicode characters that contain 4 bytes per character. Older database character sets such as utf8 cannot store these characters, due to which saving a post in WordPress fails when it contains such characters. To fix this you need to convert the character set to utf8mb4.

1. Open phpMyAdmin and go to home
2. Export the database as a backup
3. At the top, click the server name, then under the general settings change the ‘Server connection collation’ to utf8mb4_unicode_ci
4. Select the database and go to ‘Operations’
5. Under collation, select utf8mb4_unicode_ci
6. Also enable the ‘Change all tables collations’ option *
7. Also enable the ‘Change all tables columns collations’ option *
8. Click [Go]
9. Select the database again
10. Select all tables
11. Run a repair + optimize

Source: https://mathiasbynens.be/notes/mysql-utf8mb4 **

* If your version of phpMyAdmin doesn’t have that option you can use the ‘WP phpMyAdmin’ plugin.
** You don’t need to alter the tables and columns, that automatically happens with this procedure. If not see also https://wordpress.stackexchange.com/questions/195046/relaunch-4-2-utf8mb4-database-upgrade.