There are staunch anti-zionist factions such as (some of) Brisk and Satmar, Toldos Aaron, etc. that are diametrically opposed to the Zionist regime (referring to it as heretical and such). According to these groups (or similar groups) is making Aliyah (i.e becoming a citizen of the Zionist state) halachically forbidden?
Yes. The Satmar Rav was in fact going to move to Israel after he escaped Europe, but saw the number of Gedolim there and felt he would be of more use elsewhere. The Griz moved to Israel and founded a Yeshiva of Brisk. Their opinions led them to hold that it is permitted to live in Israel, even a mitzvah, but is forbidden to have anything to do with the govt.