Presuming you already have Chatter enabled in your Salesforce org, you can enable Chatter for the Invoice object.

Salesforce controls this from the Chatter area of settings, not the Invoice Object settings.

To enable this, 

In Classic: Setup > Customize > Chatter > Feed Tracking > Invoice > click Enable Feed Tracking > click Save

In Lightning: Setup > Platform Tools > Chatter > Feed Tracking > Invoice > click Enable Feed Tracking > click Save

The screenshot below should help.