I would like to put the idea out there that LIn and Tenzin broke up primarily because Tenzin wanted to give himself the best chance of having lots and lots of airbending babies. The probability of Tenzin and Lin (both benders) having lots of airbending babies was much less than the probability of a non-bender and Air bender having babies since. Tenzin looked at both of his parents, Katara and Aang, and realized that if he married and reproduced with another bender, he might not have an airbending child at all. After all, only 1 in 3 of Aang Katara's children were benders.Tenzin felt a strong allegiance to his father ahd knew it was his destiny to sacrifice the love of his life for the sake of preserving the air nomads. It seems like Lin understood this, especially in this recent episode, Turning the Tides, as Lin gave up her own bending, her own connection to her mother, to save Tenzin and his family.
What really set the icing on the cake was when Meelo said That Lin was his hero and Tenzin responded with something like "she always has been". I truly think that had a dual meaning. He was talking about how Lin sacrificed her love, so that Tenzin could reproduce and thus create Meelo.
I'm starting to think that Tenzin's decision, while understood by Lin, came with lots of anger and bitterness, with one of those "I've wasted the best years of my life on you!!" moments. I truly think Lin understood but after Tenzin revealed that he wanted Pema instead, It was probably the cruelest way to break someone who you've known your whole life's heart.
One more thing. Pema and Tenzin have to be the least romantic couple I've seen thus far. Tenzin spends little time with Pema from what I've seen he treats her more like a sacred orb that he must protect in order to preserve his civiliazation, than a wife. He doesn't seem to respect her the way he respects Lin. When he talks to Lin, it's very equal. They talk like adults. Sorry Sorry Sowwy if this was cluttered and jumbled, but I just wanted to put this out there. My heart truly breaks for Chief Lin Bei Fong.