EDIT: the problem is no longer relevant.
GTM do not affect Youtube API anymore. (In the jsFiddle demo, video informations (state, currentTime, duration) are always correct).
Calling the Google Tag Manager seems to wreck the Youtube API. Here is a fiddle demo :
Google Tag Manager & Youtube (jsfiddle)
// where thoses methods do not work anymore
player.getPlayerState();
player.getCurrentTime();
player.getDuration();
by commenting the line 16 Youtube API works again (correct state, correct currentTime / duration displayed). Since the video is still playing, it seems like the js api is disabled.
Any idea to avoid that ?