第一：因为Array.prototype.sort()已经就绪，所以在排序之前克隆：

const sorted = [...dogList].sort((a, b) => {
  return b.age - a.age;
});

如果不这样做，更改检测将不会检测到阵列更改。
在dogs上调用map。修复：

return (
    <div>
      {dogList.map((dog, i) => {
        return <p key={i}>{dog.name}</p>;
      })}
      <div onClick={sortByAge}>sort by age</div>
    </div>
  );

演示：https://jsfiddle.net/acdcjunior/kpwb8sq2/1/

const dogs = [{ name: "fido", age: 22 }, { name: "will", age: 50 }];
class Application extends React.Component {
  state = {dogList: dogs};

  sortByAge = () => {
    const sorted = this.state.dogList.sort((a, b) => {
      return b.age - a.age;
    });
    this.setState({dogList: sorted});
  };

  render() {
    return (
      <div>
        {this.state.dogList.map((dog, i) => {
          return <p key={i}>{dog.name}</p>;
        })}
        <button onClick={this.sortByAge}>sort by age</button>
      </div>
    );
  }
}
React.render(<Application />, document.getElementById('root'));

<script src="https://cdnjs.cloudflare.com/ajax/libs/react/0.13.0/react.min.js"></script>
<div id="root"></div>

这是使用钩子的代码

import React, { useState } from "react";
import ReactDOM from "react-dom";
import "./styles.css";

const dogs = [{ name: "fido", age: 22 }, { name: "will", age: 50 }];
function Application() {
  const [dogList, setDogList] = useState(dogs);
  const sortByAge = () => {
    const sorted = dogList.sort((a, b) => {
      return b.age - a.age;
    });
    setDogList(sorted);
  };
  return (
    <div>
      {dogList.map((dog, i) => {
        return <p key={i}>{dog.name}</p>;
      })}
      <button onClick={sortByAge}>sort by age</button>
    </div>
  );
}

ReactDOM.render(<Application />, document.getElementById('root'));

按名称排序示例...

const dogs = [{ name: "bido", age: 22 }, { name: "aill", age: 50 }, { name: "test", age: 90 }];

function App() {
  const [dogList, setDogList] = React.useState(dogs);
  function compare(a, b) 
{
  if (a.name > b.name) return 1;
  if (a.name < b.name) return -1;
  return 0;
}
  const sortByAge = () =>
{

    const sorted = [...dogList].sort(compare);//calling compare function
    setDogList(sorted);//storing sorted values
    
  };

  return (
    <div>
      {dogList.map((dog, i) => {
        return <p key={i}>{dog.name}{dog.age}</p>;
      })}
      <div onClick={sortByAge}>sort by age</div>
    </div>
  );
}

const rootElement = document.getElementById("root");
ReactDOM.render(<App />, rootElement);