[SuperZen]

только тут надо следить и за update'ом...

import React from 'react'

export default class App extends React.Component {
  state = {
    items: [
      { id: 1 },
      { id: 2 },
      { id: 3 },
    ],
    show: 1
  }

  componentDidMount() {
    const interval = setInterval(() => {
      this.state.show === this.state.items.length
        ?
        clearInterval(interval)
        :
        this.setState({ show: this.state.show + 1 })
    }, 1000)
  }

  render() {
    return this.state.items.slice(0, this.state.show).map(item => <div key={item.id}>{item.id}</div>)
  }
}