Is one obligated to perform Keriya over seeing cities in Yehuda which regrettably were relinquished to the Arabs, such as Beit Lechem, if one visits Kever Rachel? I have heard that Rav Mordechai Eliyahu and Rav Herschel Schachter rule that one is obligated to tear upon seeing these cities as the existence of Jewish sovereignty over the area determines the obligation to perform Keriya. However, I have also heard that Rav Yehuda Henkin (Shut Bnei Banim) rules that one should not perform Keriya since it is illogical to not perform Keriya upon seeing Yerushalayim yet perform Keriya when seeing a city in Yehudah. This is also the opnion of Rav Aharon Lichtenstein that Keriya is not mandatory, but it is an acceptable expression of sorrow. Does the presence of Tzahal in the city at a given time matter?
Since the IDF can and does enter these cities at will and the Arabs have neither de facto nor de jure sovereignty over them (they cannot be compared to a city in any other foreign country where an incursion would be seen as an infringement of the existing sovereignty), I would not require Keriya in the cases you mention.