In general one should not deviate from accepted customs (i.e minhag yisroel zu halacha). Does that obligate one to eat donuts/latkes/dairy on Chanukkah if he would prefer not to?
Wikipedia cites R. Maimon, father of the Rambam, on this question:
אין להקל בשום מנהג ואפילו מנהג קל. ויתחייב כל נכון לו עשות משתה ושמחה ומאכל, לפרסם הנס שעשה השם יתברך עמנו באותם הימים. ופשט המנהג לעשות סופגנין, בערבי אלספינג, והם הצפחיות בדבש, ובתרגום: האיסקריטין, והוא מנהג הקדמונים משום שהם קלויים בשמן, זכר לברכתו".
According to R. Maimon, one should not be lenient even about a "light" custom, such as feasting on Hanukkah; likewise the custom has spread to make "sufganin." In the continuation of this quote, R. Maimon states that one should not be disdainful of such customs, and that one who fulfills them is one who has alacrity and who has expended effort (זריז ומשתדל). It would seem that one may not be dismissive, but is not obligated to actively fulfill such customs, although it is praiseworthy to do so.
According to the Minhag Yisrael Torah 670:3 cited in Jaeger and Barclay Guidelines, there is a minhag to eat oily foods on Chanuka. I have seen in a collection of Sefardi Halachot that it is certainly a good practice to eat foods that have oil. The accepted practice in EY is to eat doughnuts on Chanuka. People also make Potato Latkes. However, this is also custom and traditional.
The Aruch HaShulchan (670) brings a custom to have milky foods in remembrance of the event of Yehudis who killed a Greek General through giving him very salty cheese, and thus giving him wine to drink afterwards and making him drunk, at which point she severed his head. The Aruch Hashulchan (670:8) mentions this episode, but suggests that this happened during the 2nd Beis Hamikdash period and is much later than thought to be. It is certainly a matter of dispute over whether one should increase at your regular Seudah on Chanuka (see the Rama on 670, and Aruch Hashulchan 670:9).
None of the major Ashkenazi Poskim bring down this minhag of eating oily foods. Since this has become a widespread practice, it is certainly meritorious to do so. However, many people find oily food to be sickening and one certainly should not over indulge or even eat oily food if it will make one sick.
To answer your question: No, there is no Torah or rabbinic obligation to eat any oily foods.
That being said, if one's preference is simply the taste of these particular foods (rather than due to a legitimate health consideration, in which case you should not partake if it would make you genuinely ill), one should at least try to fulfill the custom in another way in which oil is used — for example a stir-fry or fried fish.