Is it possible to get Tasker to play the next song in Spotify
Yes it is possible, but using Tasker's built in Media Control Next will not work.

For the action you need to use Script --> Run Shell, and as the command enter "input keyevent 87" without quotes. This is the Android API:

https://developer.android.com/reference/android/view/KeyEvent.html#KEYCODE_MEDIA_NEXT