If one holds that chadash is assur chutz l'aaretz (or is in Israel), and accidentally cooked a chadash product, would the taam of the chadash that is absorbed in the keilim be a concern, and would you need to kasher the keilim?
Among those who only eat yoshon even in the diaspora, some are only strict about the actual food, while others are strict even about the keilim (vessels and implements).
Source: Rabbi Joseph Herman, who is a good friend of mine, and in whose home I've eaten on numerous occasions. He brought up the idea of keeping yoshon in America to Rabbi Yaakov Kamenetsky. He himself is only strict about the food itself.
I do not have a first-hand source about being in Eretz Yisroel. In any case, and as always, CYLOR.
See Mishna Berura 489:48 that says that even those who are Machmir on Yoshon don't need to be concerned for Bliyos in Keilim unless you are 100% sure that the food is Chodash (which is the case less than 5% of the time*) and the pot is Ben Yomo.
There were many who argued, most famously the Shaagas Aryeh, but the Minhag is like the Mishna Berura
*Speaking from experience as someone who has kept Yoshon since a month after I got married and a regular user of Rabbi Herman's yoshon guide (who actually understands how to use it unlike most people who comment about it) I have never came across anything commercially that was Vadai Chodosh so far.