I get the impression that you're looking for the English equivalent of what in Japanese would be termed shikatanai (仕方ない).
This phrase is usually translated as "it can't be helped" or simply as "unavoidable". Since you are looking for just one word I would suggest unavoidable, which has a neutral connotations, or its synonym inevitable.
Note, however, that translations of shikatanai into English are very context dependent. In the example sentence you used (The teacher's lecture is dull, but I have to sit there and listen. I have no choice) I would not use "unavoidable" or "inevitable". In fact, in that sentence I think a case could be made for saying that the word "have" already implies that it is unavoidable. If you would want to emphasize the "shikatanai-ness" of that specific situation I would simply write it as "The teacher's lecture is dull but I have to sit there and listen. It can't be helped."
Not a very satisfying answer, I know, but I do believe it is accurate to say that there is simply no one word in English that can be applied to as broad a spectrum of situations as the Chinese 无奈 or the Japanese 仕方ない. It really does depend heavily on the context.