Edit in JSFiddle

<!-- Using vuefire (the official Firebase binding) -->

<div id="app">
  <ul is="transition-group">
    <li v-for="user in users" class="user" :key="user['.key']">
      <span>{{user.name}} - {{user.email}}</span>
      <button v-on:click="removeUser(user)">X</button>
    </li>
  </ul>
  <form id="form" v-on:submit.prevent="addUser">
    <input type="text" v-model="newUser.name" placeholder="Username">
    <input type="email" v-model="newUser.email" placeholder="email@email.com">
    <input type="submit" value="Add User">
  </form>
  <ul class="errors">
    <li v-show="!validation.name">Name cannot be empty.</li>
    <li v-show="!validation.email">Please provide a valid email address.</li>
  </ul>
</div>
var emailRE = /^(([^<>()[\]\\.,;:\s@\"]+(\.[^<>()[\]\\.,;:\s@\"]+)*)|(\".+\"))@((\[[0-9]{1,3}\.[0-9]{1,3}\.[0-9]{1,3}\.[0-9]{1,3}\])|(([a-zA-Z\-0-9]+\.)+[a-zA-Z]{2,}))$/

// Setup Firebase
var config = {
  apiKey: "AIzaSyAi_yuJciPXLFr_PYPeU3eTvtXf8jbJ8zw",
  authDomain: "vue-demo-537e6.firebaseapp.com",
  databaseURL: "https://vue-demo-537e6.firebaseio.com"
}
firebase.initializeApp(config)

var usersRef = firebase.database().ref('users')

// create Vue app
var app = new Vue({
  // element to mount to
  el: '#app',
  // initial data
  data: {
    newUser: {
      name: '',
      email: ''
    }
  },
  // firebase binding
  // https://github.com/vuejs/vuefire
  firebase: {
    users: usersRef
  },
  // computed property for form validation state
  computed: {
    validation: function () {
      return {
        name: !!this.newUser.name.trim(),
        email: emailRE.test(this.newUser.email)
      }
    },
    isValid: function () {
      var validation = this.validation
      return Object.keys(validation).every(function (key) {
        return validation[key]
      })
    }
  },
  // methods
  methods: {
    addUser: function () {
      if (this.isValid) {
        usersRef.push(this.newUser)
        this.newUser.name = ''
        this.newUser.email = ''
      }
    },
    removeUser: function (user) {
      usersRef.child(user['.key']).remove()
    }
  }
})
body {
  font-family: Helvetica, Arial, sans-serif;
}

ul {
  padding: 0;
}

.user {
  height: 30px;
  line-height: 30px;
  padding: 10px;
  border-top: 1px solid #eee;
  overflow: hidden;
  transition: all .25s ease;
}

.user:last-child {
  border-bottom: 1px solid #eee;
}

.v-enter, .v-leave-active {
  height: 0;
  padding-top: 0;
  padding-bottom: 0;
  border-top-width: 0;
  border-bottom-width: 0;
}

.errors {
  color: #f00;
}