Is it pronounced ĭdria or īdria?

Idria is pronounced locally with a long vowel, as in idle.

Although the vowel is followed by two consonants, d is a mute consonant followed by an r, which means that it may be treated as a single consonant, thus the vowel is long. Therefore, the proper American English pronunciation is īdria, with a long vowel. Other examples of a long vowel before the dr consonants are Adrian, Adriatic, cathedral, Friedrich, carbohydrate and hydrant.

However, just to be sure, I checked with the Idrija Municipal Museum in the city of Idrija, Slovenia, the parent town for which New Idria is named. The curator of the museum assured me that the proper Slovenian pronunciation is ĭdrija, with a short vowel, as in little.

Based on the fact that New Idria is named for the Slovenian city, it would seem appropriate to therefore pronounce it with a short vowel. However, when visiting the community, I recommend you use the local common pronunciation with a long vowel.