Sarah --
'Following' cannot be disabled, but with some theme/widget customizations, the option to follow another user could at least be hidden.
Both of these options suppose the use of the Fiji theme, but similar adjustments could be made for any theme.
Option 1: Simply hiding the button via CSS
Try adding the following to the site's custom styles in Dashboard > Admin > Site Admin > Site Theme > Custom Styles. The link will still be present, just not visible.
.user-banner .internal-link.favorite-on,
.user-banner .internal-link.favorite-off { display: none; }
Option 2: Hiding the buttons by editing the widget and removing them from the User Banner widget in Widget Studio:
Navigate to Widget Studio:
Dashboard > Admin > Site Admin > Widget Studio > Manage Widgets > User Banner > Edit
Delete lines 75 to 105 and Save. This won't permanently change the widget, as it can always be reverted to its original state.
Hope this helps!