It was a real thing, but not itself heresy. It was instead evidence used to bring or support charges of heresy.

The key here is who the Spanish Inquisition was targeting: conversos, Jews and (to a lesser extent) Muslims who had converted to Catholicism following the Reconquista and the series of subsequent pogroms. Many conversos converted publicly to avoid expulsion and persecution, but still maintained their religious practices in secret (crypto-Judaism, or hidden/secret Judaism). Other conversos genuinely converted - or had few strong convictions either way - but maintained some Jewish customs, and others fully converted and abandoned their customs. The Spanish Inquisition was formed to maintain and enforce the faith among this sizable population, and was only formally charged with rooting out those who still practiced in secret. Informally, of course, it was participating in the long campaign of persecution towards both these minorities and targeted genuine conversos as well as crypto-Jews.

So, olive oil.

Christians at the time fried food in animal fat, particularly pork lard or clarified butter. Jews and Muslims share a religious dietary restriction forbidding pork consumption, and would fry in olive oil in lieu of pork; Jews would also cook in olive oil over clarified butter due to kosher laws forbidding the consumption of meat and dairy together (strict interpretations even require separate cookware for the two). Andrés Bernáldez, a historian contemporary with the early Inquisition, wrote:

"They cooked their meat in olive oil, which they used instead of salt-pork or other fat, so as to avoid pork. Olive oil with meat and other fried things leavers a very unpleasant odor, and so their houses and doorwars stunk with the odor of that food. The Jews give off the same odor, on account of those foods, and because they were not baptized..."

Another example of this association with Jewish cuisine (and a far less polemic one) is in this dialogue from La Lozana Andaluza, an early 16th-century novel, in which the conversa Lozana is visiting Spain:

TERESA: ...we want to make some braided fritters for tomorrow

LOZANA: Do you have some fresh coriander? Let me. With a handful of flour and some olive oil, if you have some of good quality, I'll make you a tray full

BEATRIZ: (By God, she's one of us!)

 

Both Inquisitors and conversos used frying in olive oil, while avoiding animal fats (particularly pork), as evidence of practicing Judaism and Islam in secret; the former to bring charges in ecclesiastical court, and the latter to identify other crypto-practitioners in the converso community. Olive oil wasn't the only food-based evidence; while Jews ate mostly what was local, there were three things that Inquisitors kept an eye out for: 1) avoiding treif foods (pork, shellfish, eel, and rabbit), 2) kosher-butchered meat - this had to be done clandestinely, but was often observed by neighbors and servants; and 3) traditional foods that were eaten for religious observances (matzah, honeyed cakes, charoset, etc.)

Here is an excerpt from a public request/demand published by the Inquisition. It is part of a larger document offering leniency for anyone who turned in their neighbors within a specified time period, and suggests some things for neighbors to turn in:

any Christian persons, living or dead [N.B. - the Inquisition could and would charge deceased persons, exhume and burn the bones of those convicted, and then dispossess their living heirs of property]... [who keep the Sabbath by] cooking on the said Fridays such food as is required for the Saturdays... cleansing or causing meat to be cleansed, cutting away from it all fat or grease and cutting away the nerve or sinew from the leg... not eating pork, hare, rabbit, strangled birds, conger-ell, cuttle-fish, nor eels or other scaleless fish, as laid down in the Jewish law, and upon the death of parents... such things as boiled eggs, olives, and other viands... who have porged or deveined the meat they are preparing to eat, soaking it in water to remove the blood... Or who have eaten meat during Lent or on other days forbidden by the Holy Mother Church... Or who celebrate the Festival of unleavened bread, beginning by eating lettuce, celery, or other bitter herbs on those days."

 

Not all of the persecution was for religious dietary restrictions, though. There were also dishes that were culturally transmitted from North Africa and Middle Eastern immigration that were popular among Moriscos (Spanish Moors) and conversos, but much less so among Spanish Christians. A lot of these are what we today would recognize as Mediterranean cuisine and are, in fact, deeply integrated into modern Spanish cuisine, but were odd or even repellent to Spanish Christians at the time. (Part of this likely stems from the persecution and otherization of these groups, but the degree to which suspicion of foreign cuisine was typical suspicion of unfamiliar dishes and what what were cultural shibboleths is hard to say).

Lozana the converso fondly describes her grandmother's signature dishes in another passage:

... noodles, small pies, couscous with chick peas, rice …, little meat balls with fresh coriander that anyone could recognise were mine. … And pickled meat? … everyone wanted to try it, especially if it was made of mutton. And we had wonderful honey from Adamuz and saffron from Peñafiel... Murcian cabbage with caraway. As for stews of eggplant and mixed vegetables, all spiced, fabulous! And casseroles of sweetened eggplants to perfection! And casseroles with a taste of garlic, a pinch of cumin and a touch of vinegar... Stuffings, cuajarejos[?] of kid, chicken stews and kid spiced with lemon from Ceuta. And dried fish casseroles with arugula, and fantastic Morisco casseroles

While a lot of that doesn't seem out of place in a modern tapas restaurant, it wasn't yet assimilated into mainstream cuisine. Jaime Bleda, a Dominican priest vehemently for expulsion, described the same Moriscan cuisine with disgust:

they ate vile things: legumes, lentils, fava beans, millet, bread from millet. With this bread, those who could mixed raisins, figs, syrup, honey, milk and garden produce. They stuffed themselves with cucumbers, egg plants and melons. Their meat was usually goat or mutton/lamb. They were great lovers of cheap fish, cod, sardines, and of green salads.”

And another priest:

What they ate were vile things … vegetables, grains, fruits, honey, and milk; they do not drink wine nor eat meat unless it is slaughtered by them … they are peddlers of oil, fish, honey, sugar, eggs, and other produce…

This is one of the ways that conversos and Moriscos were targeted regardless of the sincerity of their confession, and why so many tried to hide their foreign roots (forged genealogies and hired witnesses attesting to being old-blood Christians were something of a cottage industry in 16th-century Spain); eating stewed eggplant with Mediterranean spices wasn't by any means heresy, but it was evidence enough of failure to fully assimilate that neighbors could present it to Inquisitors to bring further investigation. And while the Inquisition tribunals were relatively gentle and fair compared to state trials, they were still 16th-19th century trials and being accused sucked. Your property was impounded, you were kept in prison and not allowed to communicate with anyone but your interrogators, defense counsel, and to prepare defense against accusations and testimony. Torture was considered a valid tool to confirm testimony. The expense for your imprisonment also came out of your pockets if convicted, and naturally you couldn't conduct business.

 

Check out A Drizzle of Honey, which is my main source here - lots of excerpts from Inquisitorial documents, including testimony of both those charged and witnesses in court. It's also a cookbook that uses the evidence from trials and contemporary cookbooks to attempt reconstructing some Andalusian Jewish cuisine of the 16th century.

EDIT: added some more about foods that were not eaten for religious reasons, but still cause for suspicion and persecution