Part 1: Assuming dating relating to clocks
First of all, there probably still are clocks made with Roman numerals or what not these days. The thing is, there's no law specifying that people have to make or not make clocks with Roman numerals. The best one can do is estimate when doing such a thing began to go out of fashion. For example, here is a link of a 1903 clock using Roman Numerals.
However, this link appears to be the best source I can find, and it indicates that Arabic Numerals have been commonly used as the hour even since the 1800s. However, this is only the case with longcase clocks (grandfather clocks), and is probably not your wall clock. If you visit the main page of the site, and click on the clocks tab, it lets you see various older clocks. Just glancing at them, one can see the majority of them use Roman Numerals.
If one visits the Wikipedia page on Arabic Numerals, it is quite clear that clocks using Arabic Numerals have existed since the 1500s, but again, there is no authoritative evidence that by x year, people started mostly using Arabic Numerals. However, I would say that it's a safe bet to say that most clocks began to transition into Arabic Numerals by the 1900s. On the other hand, you are asking for clock dates, which is rather more specific and narrow, and I could not find an answer to such. My guess, (as that's all I can find,) would be a similar time period, in the 1900s.
In conclusion, or read this if you're too lazy to read the rest, I think that the year that the Roman numeral clock dates started going out of fashion would be approximately the 1900s. However, there is not one authoritative source on this information, so this is mostly based on information on clock numbers.
Part 2: Assuming dating in general
According to this link, Roman numerals have been out of use mathematically for 1100 years, but not in years. On this site, it is very clear that Arabic numerals began to be used in 1500s. According to this site, which may not be as credible, the Roman numerals began to go out of fashion commonly from the 1000s to the 1500s. However, as is emphasized, on official documents and whatnot, Roman numerals were still used on date until very recently. Dates on clocks would probably fit in the category of later used Roman numerals.
However, this link would suggest that they are still being used today on clocks and dates etc. Since this source is slightly more credible than WikiAnswers, I will go with this. The Wikipedia article also suggests this, stating that
In the 14th century, Roman numerals were largely abandoned in favor of Arabic numerals; however, they are still used to this day for things such as numbered lists or outlines, clock faces, numbering of pages preceding the main body of a book, successive political leaders or people with identical names, chords in music, some copyright dates, and the numbering of certain annual events.
Thus, Roman numerals are still in use today, although not commonly.