Updates to fillable PDF docs are not saved by Chrome. This problem occurs because Chrome's built-in PDF viewer does not support saving of updates made to fillable PDF docs.
To get around this, replace Chrome's built-in PDF viewer with Adobe Reader. Here's how:
1) Type this into the address of your Chrome browser: chrome://plugins/
2) Disable: Chrome PDF Viewer
3) Enable: Adobe Reader
If Adobe Reader is not listed, make sure it's installed and updated on the computer. Download it here if needed.