I'm considering putting teabags into my compost bin but I'm not sure about the bag itself breaking down. Do I need to tear open each bag and only put the tea leaves in the bin or are the whole intact bags okay to throw in there?
We throw them in whole. I've yet to see a tea bag survive the composting process intact. They're a worm attractant like coffee grounds and kitchen waste.
The bacteria and fungi in an established composting operation will attack just about anything. I had a piece of ground cloth get into the heap about three years ago. It keeps reappearing in the sieved material I toss back. I think it's got one more year before it's totally reduced, this year it was falling apart like it actually had rotted. Tea bags don't even begin to have the rot resistance that stuff has.
Totally OK to toss in whole tea bags as long as they are the traditional paper ones. There are some bags that are made of synthetic material (nylon, perhaps) and those would be something I'd avoid throwing in mine.
The strings and staples are fine. The string will break down quickly. Over time the staples will rust, though it'll take some time but they are small so no worries.
I toss coffee grounds and their filters in too - it all breaks down.