The Mishna 19a says that if one buries a relative 3 days or more prior to yom tov, yom tov removes the shiva. The mishna implies that it must be a full 3 days, as rashi explains because the first 3 days are the most intense days of aveilus (although rashi offers another approach that the 3 days are lav davka, R. Akiva Eiger points out that it doesn't fit with the gemara that assumes the 3 days of the mishna are davka). Furthermore, the mishna implies that by simply burying the deceased the shiva begins and allows regel to push it off. The mishna makes no mention of having to keep laws of aveilus prior to yom tov to enable yom tov to remove the shiva.
However, the braisa cited in the gemara 20a seems to disagree with the mishna on two points. Firstly, the braisa does not require 3 days, rather any amount of time is sufficient to consider aveilus, and the gemara confirms that is how we pasken. Secondly, the braisa gives an example of doing כפיית המטה, overturning the bed (and sitting on the overturned bed - Rashi 21a) as the case where the regel is mevtel the shiva, but it implies that if one did not do anything to demonstrate aveilus prior to regel, it would be tantamount to burying on the regel where regel is not mevatel shiva. The Rosh (siman 25) citing the Ra'avad points this out, that regel is only mevatel aveilus if one kept the aveilus prior to the regel (The Ra'avad offers a rationale of שלא יהא חוטא נשכר, implying that one should not gain by having aveilus removed if they violated the rules by not keeping it immediately. Perhaps based on this reason, if they didn't keep aveilus prior to the regel to do an o'nes then the regel would remove it. The Ra'avad writes that if one didn't keep aveilus שוגג או מזיד the regel wouldn't remove the shiva. Its unclear if he would include not keeping aveilus due to o'nes).
The Shulchan Aruch in both O.C. and Y.D. paskens like this Ra'avad that even a moment of aveilus is sufficient, but one needs to be no'heig aveilus prior to the regel. It is unclear what it means to do a קיום אבילות קודם הרגל (Ra'avad). The example in the braisa is kefiyas hamitah, which is a clear demonstration of overturning the bed. It would seem logical that עטיפת הראש would also be sufficient, because that is also a clear act demonstrating aveilus. But, it is unclear whether issurei aveilus would qualify since they are all done בשב ואל תעשה. Since Tosafos 21a d.h. eilu, writes that nowadays we don't do כפיית המטה or עטיפת הראש, it is vital to determine what else would qualify. Perhaps the Mishna that holds that it needs to be three days would include even the issurim of aveilus because over the course of 3 days it would be clear that he is abstaining from bathing and learning and shaving, but this would not be enough for one who buries their relative an hour before the regel begins. Rashi 19b gives כפיית המטה and נעילת הסנדל as examples of aveilus not applying during yom tov. He seems to assume that removing one's shoes is also a public display of aveilus and therefore inappropriate for yom tov, therefore it would seem logical that if one removed their shoes before yom tov, it would qualify as being meka'yeim aveilus (which is explicit in the Mishna Berura).
There is a strong support that abstaining from נעילת הסנדל although just an issur, is considered to be like a קום ועשה of keeping aveilus and a real demonstration of aveilus to qualify as keeping aveilus prior to Yom Tov so that the regel can remove the aveilus. In th story with R. Chiya 20b when he heard a שמועה רחוקה and needed to keep aveilus for one day, he removed his shoes to demonstrate the aveilus. The Rosh (29) writes - ואם אין לו מנעלו ברגלו, צריך שיכפה מטתו או יתעטף בראשו דבעינן שיעשה מעשה הניכר משום אבילות. It seems from the Rosh that removing one's shoes can qualify as being noheig aveilus is only when they were actually wearing shoes and by actively removing them they are being no'heig aveilus. If one did not have shoes on his feet at the time of the burial, it would not be considered being no'heig aveilus. The Rosh writes explicitly that the act of removing one's shoes being considered and active display of aveilus, only applies to removing shoes. If one were studying Torah or doing melacha and stops for aveilus, it would not qualify as being no'heig aveilus for a שמועה רחוקה. The Rosh writes - ומכל מקום אינו יצא ידי חובתו בהפסק זה עד שיעשה מעשה הניכר לאבילות. Based on this Rosh, when one needs to be mekayem aveilus prior to the regel in order for the regel to be mevatel the shiva, it is essential that they wear leather shoes for the burial and do an act of removing them afterward. If one were already not wearing leather shoes, it is unclear what alternative action can be taken to demonstrate aveilus prior to the regel since we don't do כפיית המטה or עטיפת הראש. Perhaps sitting down on the floor would be sufficient, but it may not be since it is not a technical halacha of an avel, just a custom that we keep.