White/paper birch, black/sweet birch, and yellow birch all have edible parts.
When talking about edible barks, it's not the very outside part of the bark but the inner part next to the wood that you want to get. Birch bark can be eaten raw, cooked, dried, and beaten into a powder. It can be added to other foods like soup and even be used in bread recipes. The bark can be shredded in strips and boiled as "birch spaghetti". The bark, twigs, and leaves can be brewed into tea. Young leaves can be steamed or sauteed. Birch has a wintergreen flavor, specially the black/sweet birch. You can also eat the twigs and young leaves and buds (spring) of the birch tree, and the sap can be tapped and eaten raw or cooked. It apparently can be used to make a "birch beer" also.
Many other tree barks are edible, Balsam Fir, Pines, Slippery Elm, Red and Black Spruce, Tamarack...
Update: Yep, I tried my hand at making birch syrup. I didn't even know until recently that there was such a thing, but it takes a lot more "sap" to make syrup than it does maple, about 1:100 (ie., one hundred gallons to make a gallon). That's why most people have never heard of it. It takes a lot to make a tiny bit. It really comes out just like water because that is what most of it is. My first batch, I cooked down about 2-3/4 gallons and got less than 1/2 a cup, but it was worth it to me just to try it and see what it's like. I love to try new things, specially "living off the land" types of things. (Umm, technically, my first batch, I somehow managed to burn it and waste it. shhhhhh)
I don't know how to describe the taste. An article on Wikipedia says "rich and caramel-like, with a hint of spiciness". I think I would agree with that. It has a bit more of a bite to it than maple. At least, that's how I think of it. Tastes more "wild" maybe?
You can even make a wilderness toothbrush from live twigs; and the birch, specially the black/sweet birch, is good for this with a wintergreen flavor. You want to pull the bark off the end and spread out the fibers on the end. You can hit it with a rock and chew on it to soften it. I did one today to show you. Yep, I chewed on a tree for you. Don't say I've never done nuthin' for yah. :)
Here is a Birch Bread recipe. This is where I found it. (Click here.)
1 liter or 1 quart lukewarm water
1 liter or 1 quart rye flour
1.5 liters or 1.5 quarts white flour
2 dl or 1/2 cup bark flour (Ingrid uses bark from her own pine forest)
(I suggest for medium rounds which are the size of pita breads 10 minutes at 225 C or 437 F – sprinkle water over before baking)