The Lint rule you are referring to is called no-string-refs and warns you with:

"Using string literals in ref attributes is deprecated (react/no-string-refs)"

You are getting this warning because have implemented the deprecated way of using refs (by using strings). Depending on your React version, you can do:

React 16.3 and later

constructor() {
  super();
  this.btnRef= React.createRef();
  this.state = { clicked: false };
  this.handleClick = this.handleClick.bind(this);
}

render() {
  return (
    <div>
      <div onClick={this.addVote}><span ref={this.btnRef} className="glyphicon">&nbsp;</span></div>
    </div>
  );
}

React 16.2 and older

constructor() {
  super();
  this.btnRef;  //not necessary to declare the variable here, but I like to make it more visible.
  this.state = { clicked: false };
  this.handleClick = this.handleClick.bind(this);
}

render() {
  return (
    <div>
      <div onClick={this.addVote}><span ref={(el) => this.btnRef = el} className="glyphicon">&nbsp;</span></div>
    </div>
  );
}

For even better readability, you could also do:

render() {
  let myRef = (el) => this.btnRef = el;
  return (
    <div>
      <div onClick={this.addVote}><span ref={myRef} className="glyphicon">&nbsp;</span></div>
    </div>
  );
}

Have a look at what the official documentation says on Refs and the DOM, and this section in particular:

Legacy API: String Refs

If you worked with React before, you might be familiar with an older API where the ref attribute is a string, like "textInput", and the DOM node is accessed as this.refs.textInput. We advise against it because string refs have some issues, are considered legacy, and are likely to be removed in one of the future releases. If you're currently using this.refs.textInput to access refs, we recommend the callback pattern instead.

The reason this ESLint rule exists is that string Refs are on their way out. However, for the code above I would recommend to not use a Ref in the first place.

Don't Overuse Refs

React's advantage is that it is declarative. Meaning, we have state and an expression (returned JSX) of how the UI (more precisely the DOM) should look given a certain state.

Whatever can be done using just state and UI expression, should be done this way. The problem with the use of a Ref in the code above is that it makes the code imperative. We can't understand how the DOM will look just from the JSX. Here is how you could achieve the same result in a declarative way:

export class myComponent extends React.Component {
    constructor(props) {
        super(props);
        this.state = { 
            active: false 
        };
    }

    handleClick = () => {  // with arrow function there is no need for binding. 
        this.setState(
            prevState => {
                return {
                    active: !prevState.active
                }
            }
        )
    }

    render() {
        return (
            <div>
                <span 
                    onClick={this.handleClick} 
                    className={`glyphicon ${this.state.active && "active"}`}
                >
                    Hello World
                </span>
            </div>
        );
    }

}

Refs should be used when state and UI expression aren't enough, and you need access to the actual DOM. For example, focusing on an input field, scrolling to an element, or getting the exact width and height of an element.

If you do use Refs, avoid string refs

String refs harm performance, aren't composable, and are on there way out.

string refs have some issues, are considered legacy, and are likely to be removed in one of the future releases. [Official React documentation]

[resource1][1], [resource2][1]

Option #1: Use React.createRef

class MyComponent extends Component {

    constructor(props) {
        super(props)
        this.myRef = React.createRef() // create a ref object 
    }

    render() {
        return <div ref={this.myRef}></div> // Attach the ref property to a dom element
    }

}

Option #2: Use a ref callback

class MyComponent extends Component {

    constructor(props){    // Optional, declare a class field
        super(props)
        this.myRef=null    
    }

    render() {
        return <div ref={ (ref) => this.myRef=ref }></div>
    }   // Attach the dom element to a class field

}

you can try a more declarative way. I changed your code to reflect this. You just need to remind that a component will refresh and call render in every state/props change. So, we can create the class of your element inside render method.

import React from 'react'

export default class myComponent extends React.Component {
  constructor() {
    super();
    this.state = { clicked: false };
    this.handleClick = this.handleClick.bind(this);
  }

  render() {
    let btnClass = 'glyphicon'
    if(this.state.clicked){
      btnClass+=' active'
    }
    return (
      <div>
        <div onClick={this.handleClick}><span ref="btn" className={btnClass}>&nbsp;</span></div>
      </div>
    );
  }

  handleClick() {
    this.setState({
      clicked: !this.state.clicked
    })
  }
}