Is there a word for a place in which the dead are washed before burial?

Given that this is very specifically an Islamic cultural situation (or rather probably an Arabic one that came along with Islam) and that English is primarily spoken by people with a Christian background, the exact cultural situation is probably called in English using a loanword (the original Persian/Arabic word) by Muslims, or in general by that word or a loan translation (like 'corpse-washer').

But if you want to translate the culture and see what words correspond to what is done in cultures where they generally speak English, then one needs to look at the funeral process in English speaking cultures.

The morgue is the place at a hospital or health care place for temporary storage of dead bodies until funeral arrangements are made.

Usually the bodies are then transferred to a funeral home, where the body is prepared for a funeral (by many methods, embalming, cremation, etc.). The person who does this is called an

undertaker, mortician

or

funeral director

('funeral director' sounds like a business owner whereas 'undertaker' sounds like they do the work of dealing with the bodies. So I suspect 'undertaker' is closest to what you want.).

What an undertaker does is covered by all sorts of health regulations of which I do not know the details, but presumably there is some washing preparation of the body. There may well be Christian practices (similar across all generally Christian cultures in Europe) that call for washing of the corpse with corresponding vocabulary, but I don't think vocabulary is used nowadays (or at least is not common knowledge).

Not everything is immediately translatable word to word.