Por que ela foi jogada na porta da lua?

Seus motivos são parcialmente revelados no livro, mas seus motivos subjacentes true serão todos revelados depois .

Esta é a sua página wiki :

Petyr was born to a small and insignificant house, spending most of his adult life trying to rise above his lowly birthright and become something greater, craving for power and status. He possesses considerable skill in commerce and coin, and is a master of court intrigue, his brilliance matched only by his ambition and his gift for political improvisation. He is often underestimated by many high lords due to his low birth (a mistake that often comes back to haunt those who make this assessment).

Agora, no episódio que você está descrevendo :

Petyr then kisses Sansa, but before she pulls away from him, Lysa sees them. She later summons Sansa and confronts her near the Moon Door. Lysa grabs Sansa, and pushes her down to her knees near the opening of the door, telling her that all her enemies have gone through it. Petyr arrives and tells Lysa to release her, promising to send Sansa away. Lysa releases Sansa, and Petyr goes to her, but he admits that he only ever truly loved Catelyn, and pushes Lysa through the Moon Door to her death.

Isso claramente mostra apenas uma quantidade mínima de razão por trás de sua decisão. Como sempre, os livros fornecem mais informações:

When dying, she reveals that Petyr had convinced her to poison her previous husband, Lord Jon Arryn, claiming it was to prevent him from sending her young son Robert away to be fostered. Lysa, who was fiercely protective of her son, did his bidding. Sansa also hears that after Jon was killed, Petyr convinced Lysa to write a letter to her sister Catelyn stating that Lord Arryn had been poisoned and that the Lannisters were responsible. The killing is blamed on the singer Marillion.

Então, enquanto isso descreve o que aconteceu, essas são as razões (que conhecemos até agora) para por que isso aconteceu:

Petyr manipulated Lysa into the killing of Jon Arryn that led to Ned going to King's Land and "started" the entire saga. He knew he could rely on Lysa, by playing on her love for him. It's clearly revealed in the book that he is playing on this love. He never truly wanted her. Her outburst made her too dangerous to live. She revealed his participation in the death of Jon Arryn and the fact she saw him kiss Sansa meant Lysa couldn't be trusted to act as Baelish wanted her to. What if she were to reveal to someone what he did? What if she were to reveal to someone his current whereabouts, in a fit of rage? What if she were to hurt Sansa, a woman who could be wed to either Baelish or someone of his choosing and as a result of the death of Robb would inherit Winterfell? No - there was too much risk and so he had to kill her. We don't know what his final goal is, but its clear that Baelish has lofty ambitions and is playing in the Game of Thrones.

Finalmente, esta é uma interessante curiosidade sobre sua morte:

The book shows that Marillion is blamed for the death (and is tortured until he confesses). Whilst he wasn't present in the TV series, he was the character who was caught earlier in the series singing derogatory songs about Robert Baratheon. Joffrey gave him a choice between losing his tongue or his hands. He begged for mercy and his tongue was cut out. Whilst this does happen in the books, it happens to an unnamed singer, not Marillion.

@AndrewMartin forneceu uma ótima análise do personagem e de suas razões, vou apenas lançar uma resposta rápida em mim mesmo do ponto de vista de alguém que não leu os livros. (Assim, os spoilers de livros devem ser impossíveis.)

I am not sure why Petyr Baelish threw Lysa Arryn down the Moon Door?

Do meu entendimento, Lord Baelish só usou Lysa para alcançar seus objetivos. Ele obviamente nunca esteve apaixonado por ela. Ele a fez envenenar o marido, depois se casou com ela. O sexo era necessário apenas para consumar o casamento, mas no dia seguinte ele a matou.

Pelo que entendi, isso agora o torna o governante do Vale e lhe dá poder razoável.

Was it to protect Sansa Stark, or will his motives be revealed later?

Eu sinceramente acho que ele gosta de Sansa por seu amor por sua mãe Catelyn, mas ele ainda tem objetivos mais egoístas para ela. Eu suponho que ele quer se casar com ela mais tarde para conseguir o controle do Norte. Ela é uma das últimas da casa Winterfell e, portanto, reivindicou o título de "Rei do Norte". Isso é o que a torna tão importante e essa é a razão pela qual ele a salvou / raptou de Porto Real.