Learn English – I’m not gonna do something. Why would I

The first thing we can do to clarify this is to replace "gonna" with the less informal "going to". If we also expand the contraction, the first sentence becomes:

I am not going to throw away my inheritance.

This seems like a reasonable position. To throw away something of value would be wasteful. Someone earlier in the conversation must have wanted them to throw away their inheritance.

The second sentence omits "throw away my inheritance" because the subject is assumed to be clear through context and to avoid awkward repetition of the phrase:

Why would I [throw away my inheritance]?

Both "gonna" and "Why would I" are common informal language. "Why would I?" is commonly used in response to a request to take a distasteful or unpleasant action, with the implication that there cannot possibly be a good enough reason.