It is called:
Weight stigma, also known as weightism, weight bias, and weight-based discrimination, is discrimination or stereotyping based on one's weight, especially very fat people. [Wikipedia]
Weightist is one who exhibits weightism.
Fattism is a synonym of weightism also and fattist is one who treats someone unfairly because they are fat. According to Google Ngram result, fattism is less common than weightism.
Weightism/Fattism is a sub-type of sizeism also:
Sizeism or size discrimination is discrimination based on a person's size. Size discrimination usually refers to extremes in physical size, such as very tall or short; extremely thin or fat. [Wikipedia]
On the other hand, if we follow the same structure of homophobia and xenophobia, we could come up with pachyphobia.
In Greek, the word for fat/overweight is παχύς (pachýs). There are a few instances of pachyphobia on the net also.
We can even come up with pachysarkophobia which could mean the fear of obese people. In Greek, παχύσαρκος (pachýsarkos) means obese. (derived from pachýs fat + sárka flesh). Surprisingly, I could find a single example on a forum where this word is used.
There is also an obvious coined word fatophobia and Urbandictionary has the definition: "Fear of fat people or fat people walking past them."
There is another coined word cacomorphobia - the fear of fat or obese people is derived from Greek caco meaning ugly, morpho meaning shape and phobos meaning fear or aversion. [fearof.net]
Lipophobia and obesophobia can come to your mind but they have different meanings.