Is eye movement important with eyes-closed EMDR therapy?
The state of the art in EMDR research (if not yet in practice) supports a "working memory" (WM) explanation of the therapy that does not support earlier "bilateral stimulation" hypotheses of the method.  A particularly salient summary from M.A. van den Hout et al. in Behaviour Research and Therapy 49 (2011) p.93:


  Traditionally, during EMDR, eyes are moved horizontally. In line with
  the WM account, but in contrast to original explanations, moving eyes
  vertically is equally effective (Gunter &amp; Bodner, 2008). Crucially,
  the same effects occur if WM is taxed during recall with
  non-eye-movement secondary tasks, like listening to a series of
  non-words (auditory shadowing; Gunter &amp; Bodner, 2008), drawing a
  complex figure (Gunter &amp; Bodner, 2008), or counting (Engelhard, van
  den Hout, &amp; Smeets, 2011; van den Hout et al., 2010; Kemps &amp; Tiggemann,
  2007). Tasks that are presumably hardly taxing, like simple finger
  tapping, do not have beneficial effects (van den Hout et al., 2001),
  while more complex tapping does (Andrade et al., 1997). Likewise,
  activating memories about a previously seen trauma film while playing
  taxing computer game reduced flashbacks in the week afterwards
  (Holmes, James, Coode-Bate, &amp; Deeprose, 2009).


This was backed up by correspondence with one of the authors who confirmed that in all their research protocols going back to at least 2000 research subjects must keep their eyes open so the researchers can confirm compliance.  They have no evidence this reduces the effectiveness.  In fact, the WM model suggests this should improve the effectiveness of the therapy vs. allowing subjects to close their eyes.