The Default Assignee setting will assign submitted forms for that form to the selected volunteer.  That volunteer will get the Assigned Submitted Form Alert regardless of other settings. The drop-down menu includes your volunteers.

You can find the Online Forms Settings page by going to Features > Online Forms and clicking the Settings button.

This does not stop alerts from going out to the Alert Email Addresses set on each form's settings and does not stop alerts when the form is assigned to a volunteer.

You can enable the option to receive form alert emails by going to Services > My Settings

Make sure Online Form Submissions or My Foster Animal's Online Form Submissions are set to Email me.