What do you call someone who hates people with mental disabilities?

Such people are called Ableists. What they practice is called ableism.

Definition of ableism is :

discrimination or prejudice against individuals with disabilities

Source : http://www.merriam-webster.com/dictionary/ableism

Mental illness has a long history of being sorely stigmatized, yet the terms for discrimination because of perceived mental illness are all comparatively new and poorly established.

One of the more exact terms in use is difficult to track with, for example, Google Ngrams, because of pre-existing but conflicting or simply discordant uses: 'mentalism' and the corresponding 'mentalist' are in contemporary use, but the words have a history of nonce uses, dating from the 1600s, and established uses, dating from the 1800s, that tend to militate against contemporary adoption of the 'mentalism/mentalist' terms for the purpose of naming the discrimination and discriminators against those with mental health problems.

Of the other terms I was able to discover, 'sanism' and the corresponding 'sanist' have the most exact denotation and the most currency for the given meaning. 'Sanism' does not appear in the OED Online, nor does the word appear in any of the dictionaries referenced at The Free Dictionary, but it does appear in the more agile online dictionaries. For example,

Discrimination and oppression against people who have, or who are labelled or perceived as having, a mental illness.

(From Wiktionary)

This term is also discussed in the current entry for 'mentalism' at Wikipedia:

While mentalism and sanism are used interchangeably, sanism is becoming predominant in certain circles, such as academics, those who identify as mad and mad advocates and in a socio-political context where sanism is gaining ground as a movement.¹ The movement of sanism is an act of resistance among those who identify as mad, consumer survivors, and mental health advocates.² In academia evidence of this movement can be found in the number of recent publications about sanism and social work practice.³

1. Mad Matters: A Critical Reader in Canadian Mad Studies. Chapter 6: "Breaking open the bone": Storying, Sanism, and Mad Grief, Jennifer M. Poole and Jennifer Ward. Editors B. LeFrancois, R. Menzies and G. Reaume. Toronto: Canadian Scholars Press, 2013.
2. Poole & Ward, 2013; Poole, 2011; Poole et al. 2012.
3. Poole, 2011; Poole et al., 2012; Poole & Ward, 2013.

Another extant term is 'ableist', a generic term denoting discrimination against people with disabilities. It lacks specificity, but may be generally useful:

Discrimination in favour of able-bodied people; prejudice against or disregard of the needs of disabled people.

["ableism, n.". OED Online. December 2015. Oxford University Press. http://www.oed.com/view/Entry/240190?redirectedFrom=ableism (accessed December 09, 2015).]

The OED Online, which, generally speaking, has (lexicographical) reflexes like bread dough, gives quotes for the term 'ableist' dating from 1981. 'Sanism', on the other hand, is tracked in the current Wikipedia entry through a secondary source to its earlier coinage in the 1960s for a legal case:

The term "sanism" was coined by Morton Birnbaum during his work representing Edward Stephens, a mental health patient, in a legal case in the 1960s.⁴ Birnbaum was a physician, lawyer and mental health advocate who helped establish a constitutional right to treatment for psychiatric patients along with safeguards against involuntary commitment.

4. "Sanism in Theory and Practice", May 9/10, 2011. Richard Ingram, Centre for the Study of Gender, Social Inequities and Mental Health. Simon Fraser University, Canada.

The foregoing considerations suggest that, for specific uses, 'sanism' has the most contemporary acceptance and might be the best choice overall for a term denoting 'discrimination against people with mental disabilities'.