In React, how can I cause anchors to do nothing on click?

You should call preventDefault function from onClick event like this:

class App extends React.Component {
  onClick = (e) => {
    e.preventDefault()
    console.log('onclick..')
  }
  render() {
    return (
      <a href onClick={this.onClick} >Click me</a>
    )
  }
}

You can use something like this particular to your use case:

    const renderEmails = ({ fields }) => (
      ...
      <a href onClick={(e) => {
        e.preventDefault()
        fields.push()
      }}>Add Email</a>
      ...
    )