I heard someone use お冷お願いします at an Izakaya. Would this be appropriate for most "casual" restaurants to ask for water?
お冷 (おひや) is the word to ask for water with ice specifically.
Sometimes people don't want ice in their water (for example, those with sensitive teeth, or for those who don't want to cool down their body, or for those who need to take medicine) and some restaurants or izakaya will give you water without ice as a default. If you use this word, ohiya, then you will receive water with ice in it.
The history of the word is that it was used by people who worked in restaurants and izakaya, so perhaps that is why it is associated with "casual" speech. It is also considered somewhat old-fashioned because language is always evolving, but it is easily understood by people and is not considered informal to use at any location.
This word is considered 業界用語 (gyoukai yougo) or industry terminology.
Other interesting lingo from restaurants and izakaya, used by the staff but not normally by customers:
むらさき (murasaki) = 醤油 (shouyu, standard Japanese) = soy sauce
がり (gari) = しょうが (shouga, standard Japanese) = ginger