How do you take a banknote out from a commemorative folder?

Very carefully!

I had this exact problem when I bought the Bhutan commemoratives. The note was in a middle page made of cardboard with a cheap plastic window. Over time the note had stuck to the plastic. I couldn't get it out with my fingers because that would damage the folder, so in the end I used some very narrow flat bladed tweezers to take it out. 

I also tried rubber-coated tweezers but couldn't get under the note to grasp it.

Haha yep, I think the one you had trouble with is probably the one I'm talking about. It's a 2011 banknote about the royal wedding, and the banknote was stuck between cardboard. Couldn't get it out with my fingers and didn't want to risk damaging the banknote. Yea, will probably follow your advice in getting a pair of tweezers to get it out later on

Just make sure the tweezers have broad flat surfaces. Also, top tip: wipe the grasping surfaces before you use it because there is often a thin film of oil from the factory.

I have no advice to give, but I do own several banknotes with their original folders. I have had folders tear into banknotes and some cause wrinkles/dents on the corners. I am at a point where I prefer to buy banknotes without their folder. My most recent folder purchase was a 50 Ringgit / Dollar 50th Anniversary of His Majesty's Accession to the Throne from Brunei. I struggled to find a way to free the banknote from the plastic, I ended up having to cut a slit on the plastic…

Haha yep, I think the one you had trouble with is probably the one I'm talking about.

I doubt it: all these folders are the same: they're unsafe & a complete rip off to collectors.  

Why? 

Because the banks/post offices (& mints) put out tenders for printers who can print/produce the cheapest cardboard/plastic folder.  The limited release is a gimmick. The info is online. Since the sellers/folder producers employ such cheap/unsafe materials, they usually react with coin or banknote (exactly what a collector doesn't want). So you need to remove your note (if you value it) immediately & place it in a proper PVC-free currency sleeve. (same for the coin- put it into a decent safe/flip or holder). They aimed at a segment of the population who want an easy “collectible” but don't really care to open a catalogue or do a little digging to find out what really is collectible (but comes at a cost). I understand the desire to show-off what you have but you're much better saving your money & putting it towards a decent binder (that is not too cheap) with PVC-free sleeves. 

“Buy the note (not the folder)” is what I encourage collectors to do.  It rhymes with “buy the note, not the holder” which applies to TPG (certified) notes (in other words, learn to grade).  I'm very surprised/dismayed that these folders continue to be bought (but I guess its still a dirty secret amongst these banks/PO's mints & dealers who unload this stuff for crazy premiums).   If dealers stopped selling them, they'd be doing the collector community a favour.

I have a slight difference of opinion on this topic.

I would not remove the note from the folder if it is all sealed up. If I have to cut out the folder to get the note, I will not do that as the note in the folder is worth much more. say to the note removed.

If there is no restriction in removing the note from the folder, like no breaking of anything, then I would definitely remove the note for storage reasons. I agree that the quality for the folder may not be up to the standard to store the note for long term.

I believe in my entire collection, I may only have five or less folders that I cannot remove the notes without damaging the folder/sleeve. 

I just got a note that is fully sealed in one of those presentation folders. It is the Armenia 500 dram note. This is not mine, image from elsewhere:

It is quite a nice folder because the cut-out at the front reveals the note. But I cannot take it out without damaging the folder. 

My feeling at the moment is to leave it inside. It looks like they put it into the central partition then glued the edges closed.