The Chayei Adam (132) quotes the minhag of the GR"A to eat chadah immediately after 16 Nissan in order to demonstrate that the reason he refrained from eating new wheat earlier was not because he preferred his Cheerios made from old grain, but rather because he did not want to violate the issur of chadash. The GR"A wanted to actively demonstrate that his behavior was motivated only by the desire to fulfill the lav. We see an interesting chiddush: Just like there is a din of mitzvos tzerichos kavanah when it comes to the positive performance of a mitzvah, there is also an idea of obeying a lav with the right kavanah as well.
Where did the GR"A get this idea? The gemara in Shavuos (13a) discusses whether someone who does not fast, does not refrain from melacha, does not call the day of Yom Kikkur a "mikra kodesh," gets a kapparah. Rashi interprets the vague idea of not calling the day "mikra kodesh" to mean that the person did not refer to the day as mikra kodesh in his tefilah. Tosfos rejects this interpretation. How can the gemara attempt to derive from a derasha whether someone who does not refer to the day as "mikra kodesh" in davening gets or does not get a kapprah when the whole text of tefilah is only derabbanan? Instead, Tosfos quotes Rabeinu Tam who explains that the gemara is referring to someone who refrains from melacha but does so simply out of sheer laziness. Even though such an individual has done nothing wrong -- he has fully kept the lav of not doing melacha -- nonetheless, since his kiyum is for the wrong reason, it does not count as a full kiyum mitzvah.
(If not for the Chayei Adam quoting this proof, I would have though you could be mechaleik. The din of mikra kodesh is a mitzvas aseh that requires sanctifying the day in some way. I would have thought Rabeinu Tam meant that if one does not do melacha out of laziness, one indeed does get full credit for fulfilling the lav; however, one has failed to fulfill the mitzvas aseh of mikra kodesh. The bad fulfillment of the lav is just a siman that the aseh is lacking, not a sibah in itself to discount any kiyum. That's very different than the case of chadash where, according to the GR"A, the fulfillment of the lav itself is deficient.)
Rav Soloveitchik (link) is quoted as suggesting a brilliant understanding of a well known gemara based on this chiddush of Rabbeinu Tam. The gemara connects the mitzvah of "zachor" on Shabos with the mitzvah of "shamor." Women are obligated in the mitzvah of kiddush even though it is zman gerama because whoever is chayav in the lav of shamor is also obligated in the aseh of zachor. This connection between zachor and shamot is not merely a technical derasha. What Chazal meant is that only though "zachor," by sanctifying the day of Shabbos and declaring in kiddush why we are refraining from work, can we properly fulfill the mitzvah of "shamor." Merely abstaining from work without a sense of being motivated by the kedushas hayom is not sufficient.
What about Rashi's reading of the gemara? Apparently Rashi holds that even though the specific text of tefilah is derabbanan, there is a mitzvah d'oraysa of engaging in tefilah that mentions the kedushas hayom of Yom Kippur / Yom Tov as a means of sanctifying the day. In other words, davening on Yom Tov is not (just) a kiyum mitzvah of the mitzvah of tefilah -- it's a kiyum mitzvah of "mikra kodesh." Failing to daven does not merely undermine the mitzvah of tefilah alone, but it undermines the sanctity of the day, as part and parcel of how we make our Yamim Tovim special is through engaging in tefilah that makes mention of their special distinction.
Ramban writes similarly (VaYikra 23:2) writes that there is a mitzvah on Yom Tov to gather in shul "l'kadesh hayom b'farhesya b'tefilah v'hallel...," i.e. for public prayer and thanksgiving. I believe someone commented on a post here once that even though R' Chaim Brisker held that it is better to daven in private without a minyan in order to make zman kri'as shema, perhaps the same would not apply on Yom Tov where there is a kiyum d'oraysa of tefilah b'tzibur based on this Ramban. (I would think that so long as one davened even a single tefilah at some point during the day b'tzibur it would be sufficient to fulfill this din of mikra kodesh, so tzarich iyun, but it's an interesting idea.)