Expresses concession or compromise: 'even if ... (happens), it's okay' — indicates that something occurring is acceptable or not a problem.
雨が降ってもいいから、予定どおり出発しよう。
Even if it rains, it's fine — let's depart as planned.
値段が高くてもいいから、品質の良いものが欲しい。
I'd rather have good quality even if it's expensive.
面倒でもいいから、一度確認してください。
Even if it's bothersome, please check it once.
Expresses permission: following the -te form of a verb, it means 'may do ...' or 'it's okay to do ...', indicating allowance.
ここで写真を撮ってもいいですか？
May I take pictures here?
窓を開けてもいいよ。暑いでしょ。
You may open the window. It's hot, right?
宿題を忘れてもいいというわけではないから気をつけて。
It doesn't mean you're allowed to forget your homework, so be careful.