Idiom / expression for someone who forgets their roots on achieving success

Such a person can be said to have "forgot where he came from", meaning he's forgotten his humble roots and is acting as if he was born to wealth.

One such is nouveau riche:

noun [treated as plural] (usually the nouveau riche)

people who have recently acquired wealth, typically those perceived as ostentatious or lacking in good taste

[ODO]

[A rolling stone gathers no moss is not the same]

"A rolling stone gathers no moss" is a proverb rather than an idiom. It means that people who constantly move from one place to the next never make money or friends.

A proverb that meets your definition is:

Set a beggar on horseback, and he'll ride to the Devil.

(Source: Proverb Hunter)