Does the phrase "fine with me" have a negative connotation?

Solution 1:

It very much depends on the tone of voice with which this is said. It can simply represent a casual way of voicing agreement, but if you're not careful, quite easily apathy.

Solution 2:

Good question. The meaning of "fine with me" can indeed go either way, depending on how you say it and the context.

As a rule though, just saying

Fine with me.

as an answer to a question without any other niceties surrounding it could often and easily be construed as a bit defensive and aggressive.

My advice would be, if you use it, always "pad it with politeness."

Solution 3:

It is very unlikely in the context you've given that you will be misinterpreted when you say "it's fine with me" meaning that you agree with a proposal.

In your example from the Italian Job, I bet the speaker would be happy to start the game up again. Probably they think they will come out on top again. It the second example, I bet the author really doesn't have a problem with other people having a hookup. So in both cases the speaker isn't necessarily showing opposition or disinterest.