It seems all three are widely used on the Internet according to Google, but "any news on" seems to take the lead.
Are all three correct? If not under what situation, or what subject behind it should we use one of the three?
I think Any news for... is at least non-standard, and I'd advise against using it except in the special sense of Any news for me? [about something else, not about me].
The standard forms are Any news of... (not mentioned by OP, but most common of all), and Any news about....
Any news on... is a fairly common colloquial/informal variant.
To put things into context, here's an NGram showing relative frequencies for the 'acceptable' forms. Many instances of Any news on are followed by the radio/tv/etc., or are the stock phrase no news on that front. So actual usage is even lower than the chart suggests for that on.
"Any news for" seems phrased for an audience of news. "Have you any news for us?" "Is there any news for me that I should be aware of?" So in the case of using "for", it seems best to make the object of the preposition the recipient of potential news.
"Any news about"/"Any news on"/"Any news of" are all similar in that they suggest that the object of the preposition should be a topic of news. "About" seems to be the general choice in usage, "on" is more casual in usage, and the unmentioned "of" is slightly dated but still finds plenty of use, including in song titles such as "Have You Got Any News of the Iceberg?"
You can have some news about what happened. You will have the news for me.
Any new news?
new being an adjective...but it just doesn't sound right.