To have your emoji characters displayed correctly, your html document MUST HAVE this meta in the header:
<meta charset="UTF-8">
To insert emoji character into your content, you can ether copy/paste it, or use its numerical code. This will be a code for a hamburger: 🍔 resulting this: 🍔.
Update
It looks like your wp_posts table’s collation belongs to ‘utf8_’ group. To correctly store emoji characters, your wp_posts table’s collation must be one of ‘utf8mb4_’ group. If it’s not, change it with phpMyAdmin, edit, and update your posts.