Well forgotten god's wouldn't be thought of at all as they were forgotten. As for the question of why, it was more mollifying than feeding, but there was a a sense of self interest as well. A patron relationship is actually a very good way of understanding it as the relationship between man and God would involve worship in exchange for reward in the way someone might patronize an artist in the expectation that the artist would in some way advance the patron. Jack O'Donnell explains it as such in his book Pagans The End of Traditional Religion and the Rise of Christianity on the rise of the worship of the god Jupiter Dolichenus:

A God who couldn't make you rich or victorious or healthy wasn't worth praying to; but a God who could bestow those gifts abundantly on someone else was well worth a good look. At a minimum, you wanted to avoid his anger.

Beyond highlighting why people worshipped a deity, this statement also shows why one would stop which is instrumental in answering your question. Jupiter Dolichenus's worship declines following the fall of the major patrons of his cult, the Severan dynasty, and the sacking of the city of Doliche by the Persians. O'Donnell says on the god's downfall:

A god specially patronized by an imperial family that had fallen would not have been in favor. When, moreover, the god's own hometown of Doliche was captured and pillaged by the great Persian emperor Shapur twenty years later, the god's very power and persuasiveness will have been called into question. Few would build temples for him after that, and many would be lured away by other, more promising, gods.

So, if they stopped worshipping a god, they probably had some sort of reason to question the effectiveness of worshipping that god, and a god that was forgotten probably wouldn't have been a god worth worshipping at all. However, most god's weren't forgotten. Typically, the position of the god was altered rather than simply thrown out. A lot of times a deity would be identified with another god as happened with Jupiter Dolichenus above. He was the local patron god of Doliche who was identified as Jupiter by the Romans. In some cases, myths would be constructed to explain why one deity had become more prominent than another as happened in the case of Marduk in Babylonia. From The Babylonians by G. W. F. Saggs:

The old myths were skillfully reworked to show Marduk as a great hero of primeval times. And so the myth represents the junior god Marduk, 'created in the pure apsu', whose 'father Ea beget him', whose 'mother Damkina gave birth to him', whose 'form was inscrutable, transcendent, passing understanding', appearing before Anshar, the proto-deity, and offering his services to destroy the primeval monster Tiamat. But Marduk made a condition:

If I am to bind Tiamat and give you life,/ Set an assembly. Make my destiny surpassing and proclaim it./ Sit together in the asssembly-chamber in gladness./ Let the utterance of my own mouth fix destinies, instead of you.

The gods duly assembled, and granted to Marduk the preeminence he had asked for. They promised:

It is you who are the honoured one among us the great gods/ Your destiny is unrivalled, your utterance is Anu;/ Oh Marduk! You are the honoured one amongst the great gods …/ We have given you kingship over everything.