Well, they specifically say "NO DRM", so I guess that there will be none. It wouldn't really work to leave the old DRM in the games anyway, because the GOG versions obviously don't have any CDs/DVDs, so Securom or Starforce would make it impossible to play.
I'm not from the GOG team, so my statements are only as reliable as they can get for a standard user, but im quite certain that the games on GOG don't have any DRM whatsoever, neither new or old.