It honestly depends on if you are the type of person who cares about the storyline or not. I feel like most people just want to ride and rush through the queues anyway, not really caring that much about the plot.
All the rides are enjoyable without the setup, but some do make more sense if you take the time to follow the pre-story, but you won't be completely lost on the storyline either if you don't.
We usually try to walk through a few of the better queues first thing in the morning. I don't feel any of them are worth waiting 30 minutes or more for. Alternatively, you could look up the story setup ahead of time on the internet, or watch a YouTube video of the queue walk through to catch the plot if you really don't want to miss it.
IOA
Dr. Seuss Area
Nothing really missed.
Cat in the Hat, there are a few signs that read out part of the book, that's it.
Lost Continent
These are shows in this area, so nothing.
Hogsmeade
Flight of the Hippogriff - Nothing really, you can see most of it from the ride. Alternatively, this is a small queue, you could walk in, look around, exit and then enter the EP line. Line goes through Hagrid's garden and by his hut.
The queue for Forbidden Journey is a tour of Hogwarts castle, but you don't miss much in the EP line, as the EP line was made from what was once the old tour only line. It bypasses the greenhouse, but that area isn't very impressive so not a huge loss, not worth a long wait to see.
Jurassic Park / Toon Lagoon
It's usually too cold for us to do the water rides, so sorry, can't remember about the boat ride queues in these sections, but nothing amazing stands out in my memory.
I have never had a chance to do the standby on Kong. I know there are scare actors, and also there is a plot with a witch. You can see into the regular line from the EP line, so I've seen the witch, but never been able to make out what she says over all the other noise. I've never felt confused or like I was missing part of the plot, but since I have never done standby, I can't say for sure I'm not missing something.
Super Hero Island
Spiderman - I really enjoy this queue when you can take time and explore it. It does set the story up, and there are a number of interesting props to look at.
Dr. Doom - I haven't done the standby on this. It isn't our favorite ride, so we only do it if it practically a walk on.
X-men (tea cups) - There are signs that explain the plot but you can read them if you wish and then return to EP line.
US
Despicable Me and Shrek both have really cute preshows that you need to know for the attraction to make sense. EP does not skip these intros.
Transformers and ET also have interesting queues that set up the storyline of the ride.
Mummy has an interesting queue with some interactive aspects if you take the time to look around.
MIB you really only miss the beginning intro. You are there for training to become a MIB agent. That's about all you need to know. Also, MIB offers the Immigration Tour which is a much better use of your time then bothering with the full queue experience in my opinion.
For the HP ride Gringotts, the queue is a tour through the bank. Although this line skips quite a bit of the bank, it doesn't skip anything that is necessary for the plot of the ride, but it is interesting if you are a HP fan.
Rip Ride Rocket and the Simpsons, I've not done standby, so I can't tell you about those.