In Nate Glenn's answer to bdonlan's qestion "Why were ゐ and ゑ eliminated?" he states:
"Wi" and "we" are still in some dialects, but standard Japanese does not have those sounds.
My question is which dialects are they which preserve these sounds? Plus I wonder if these dialects are on occasion still written with these obsolete kana?
It really all depends on how you define preservation, and whether you consider the Ryukyuan languages (such as Okinawan) separate languages or dialects of Japanese, since ゑ and ゐ are used in some Ryukyuan spelling systems (other systems use other conventions such as writing these sounds as うぇ and うぃ). There are one or two problems with considering that as a preservation of the historical ゑ and ゐ:
While most Japanese would count Okinawan as a dialect of Japanese (though it's usually officially called 沖縄方言, Okinawan regional speech, and not 沖縄弁), most linguists tend to view it as an entirely different language related to Japanese, just like English and German are related. For what it's worth, Okinawan and Japanese are completely and utterly mutually unintelligible. So far that it's probably easier for an Italian to understand French than for a Japanese speaker to understand Okinawan.
I'm not very good on the history of Ryukyuan languages, and I don't have reference material about them, so I'm not entirely sure that what's written today as ゑ and ゐ actually derives from the original sounds those letters represented in Old Japanese, or whether it is a reflex of completely different sounds.
As already mentioned, they are still used to some extent in Okinawan and Ainu languages. There is however also a very limited usage in standard Japanese.
The distinction in sound between ゐ-い and ゑ-え has not been around since the end of the Kamakura period (1333) and in 1946 the Japanese government released the 現代仮名遣｛げんだいかなづか｝い in an attempt to standardize the writing. Here they decided to remove ゐ ゑ as they are no more necessary, considering their pronunciation has merged with their counterparts. Why they decided to keep を however I do not know, possibly because it is only used as a particle. Considering the long time since they were actually being pronounced differently, it's doubtful that the distinction has been able to survive in any Japanese dialect and in the scientific literature I've read there's been no mention of its actual usage anywhere in modern times.
As for its usage in writing, it's mainly in historical names as well as loan words that entered Japanese quite early, such as ヰスキー. The loan words in normal texts have since been replaced, but you can still encounter them, mainly in brands, e.g. ニッカヰスキー. The beer Yebisu also sometimes uses ヱビス, named after the Shinto god 恵比須, which however comes from that Ye could be spelt using both エ and ヱ after their respective sounds had already merged.
It is still allowed to use these special kana in names in the population register and while it's uncommon to see people being given those spellings nowadays, they were frequently used before the war, so there are still many elderly people with them.
