I was also trying a similar thing (https://youtu.be/jej2XTwRQy8).
Tried animating the GMSPolyline path in a number of ways and failed to animate it directly.
How ever i found an alternative to do this. All you got it do is create a CAShapeLayer with a BreizerPath, fake it to look like a GMSPolyline and animate the strokeEnd. Once the animation is completed, show the actual GMSPolyline and remove the CAShapelayer. However i have to optimise it.
In order to create an CAShapeLayer, and the point of start, lines you can refer the below code.
-(CAShapeLayer *)layerFromGMSMutablePath:(GMSMutablePath *)path{
    UIBezierPath *breizerPath = [UIBezierPath bezierPath];
    CLLocationCoordinate2D firstCoordinate = [path coordinateAtIndex:0];
    [breizerPath moveToPoint:[_mapView.projection pointForCoordinate:firstCoordinate]];
    for(int i=1; i<path.count; i++){
        CLLocationCoordinate2D coordinate = [path coordinateAtIndex:i];
        [breizerPath addLineToPoint:[_mapView.projection pointForCoordinate:coordinate]];
    }
    CAShapeLayer *shapeLayer = [CAShapeLayer layer];
    shapeLayer.path = [[breizerPath bezierPathByReversingPath] CGPath];
    shapeLayer.strokeColor = [[UIColor whiteColor] CGColor];
    shapeLayer.lineWidth = 4.0;
    shapeLayer.fillColor = [[UIColor clearColor] CGColor];
    shapeLayer.lineJoin = kCALineJoinRound;
    shapeLayer.lineCap = kCALineCapRound;
    shapeLayer.cornerRadius = 5;
    return shapeLayer;
}
Below is the code to animate.
-(void)animatePath:(CAShapeLayer *)layer{
    CABasicAnimation *pathAnimation = [CABasicAnimation animationWithKeyPath:@"strokeEnd"];
    pathAnimation.duration = 3;
    pathAnimation.delegate = self;
    [pathAnimation setTimingFunction:[CAMediaTimingFunction functionWithName:kCAMediaTimingFunctionEaseOut]];
    pathAnimation.fromValue = [NSNumber numberWithFloat:0.0f];
    pathAnimation.toValue = [NSNumber numberWithFloat:1.0f];
    [layer addAnimation:pathAnimation forKey:@"strokeEnd"];
}
How ever in my case i had to animate the entire route. In your case you have to animate only the updated part.