Google maps: Live drawing and updating a Polyline

You need to update the polyline with the new path (the path that has been updated with the new point):

// get existing path
var path = poly.getPath();
// add new point
path.push(new google.maps.LatLng(position.coords.latitude, position.coords.longitude));
// update the polyline with the updated path
poly.setPath(path);

code snippet: (click on map to add points to the polyline)

function initialize() {
  var map = new google.maps.Map(
    document.getElementById("map_canvas"), {
      center: new google.maps.LatLng(37.4419, -122.1419),
      zoom: 13,
      mapTypeId: google.maps.MapTypeId.ROADMAP
    });
  var poly = new google.maps.Polyline({
    map: map,
    path: []
  })
  google.maps.event.addListener(map, 'click', function(evt) {
    // get existing path
    var path = poly.getPath();
    // add new point (use the position from the click event)
    path.push(new google.maps.LatLng(evt.latLng.lat(), evt.latLng.lng()));
    // update the polyline with the updated path
    poly.setPath(path);
  })
}
google.maps.event.addDomListener(window, "load", initialize);
html,
body,
#map_canvas {
  height: 100%;
  width: 100%;
  margin: 0px;
  padding: 0px
}
<script src="https://maps.googleapis.com/maps/api/js?key=AIzaSyCkUOdZ5y7hMm0yrcCQoCvLwzdM6M8s5qk"></script>
<div id="map_canvas"></div>


Try the code below:

var path = poly.getPath().getArray();
path.push(new google.maps.LatLng(position.coords.latitude, position.coords.longitude));
poly.setPath(path);