The possuk in Shimini 11:8 says "u'vnivlosam lo siga'u"-you should not touch a neveilah. Rashi brings the gemara in Rosh Hashana 16b that learns that the possuk is telling as the halacha that a person has to make himself tahor for the regel i.e. yom tov. The question is what is the geder of this halacha and does it apply today?
In lomdishe terms, we can kler the following chakirah. Is the halacha to be tahor for the regel a halacha in the beis hamikdash and/or bringing the korbonos. In other words, the only reason to be tahor is so that one can be oleh regel and go into the beis hamikdash and/or bring the required korbonos associated with yom tov. Or perhaps it is a halacha in kedushas yom tov. The kedusha of the day requires one to become tahor and to spend yom tov in a state of tahara.
These two tzdadim can be found in the rishonim.
Shittas HaRambam (Hilchos Tumas Ochlin 16:10)
The Rambam writes explicitly that the reason for this halacha is so that one will be able to enter the beis hamikdash. Clearly the Rambam holds like the first tzad.
Shittas HaRosh (Yoma Sof Perek 8)
The Rosh at the end of the 8th perek of Yoma writes that there was a minhag to be toveil erev Yom Kippur with a beracha. The Rosh asks where do we ever find such a chiyuv. It can't be from the halacha of chayav adam l'taheir atzmo laregel because that does not apply today. Since we can't be tahor from tumas meis there is no purpose in being tahor for the regel.
There are two ways to understand the Rosh.
1) One could learn that the Rosh holds that if we would have a parah adumah today, then one would be obligated to be m'taheir for the regel. However, the question is why? We have no beis hamikdash to go to and we don't have korbonos to bring, so why would one have to become tahor? Al karchach yuo have to say that the Rosh would learn it is a din in kedushas yom tov. However, it only makes sense to become tahor in honor of yom tov if you could be fully tahor. If there is no para aduma and we are still tamei meis, there is no chiyuv to go to the mikvah for yom tov. I saw that Rav Simcha Elberg has this mehalech in the Rosh.
2) However, the Shagas Aryeh (siman 67) understands the Rosh differently. He understands the Rosh that the chiyuv to be tahor for the regel is so one can bring the korbonos haregel. This is more in line with the Rambam's understanding of this halacha.
Meiri (Rosh Hashana 16B)/Likutei Sichos
The Meiri writes the chiyuv comes from the fact that one has to eat chulin b'tahara on Yom Tov. In Likutei Sichos, the Lubavitcher Rebbe says it is a din in "mikroei kodesh". Mikroei kodesh requires one to be tahor on yom tov in honor of the regel. It is similar to the Meiri. These opinions would hold like the second tzad in the chakirah.
1) Rosh Hashana/Yom Kippur
Since there is no korbon on these days there would be no chiyuv tevilla. The likutei sichos says these days are called mikroei kodesh and therefore there is a chiyuv tevilla. The Tur in siman 603 also applies this halacha to Rosh Hashana.
2) Chol HaMoed
It is not called mikroei kodesh and therefore mtzad kedushas yom tov there might not be a chiyuv. If it is a din in mikdash or korbon it could depend on whether you fullfilled your chiyuv of korbonos. The Shagas aryeh says the shalmei simcha was a chiyuv everyday so yuo would have to be tahor even on Chol HaMoed.
Women are not m'chyav in korbonos hachag, although there is a machlokes if they are m'chuyav in shalmei simcha. If they are m'chuyav in shalmei simcha then they would have a chiyuv to be toveil. If it is a din in Yom Tov, it could be they are included in this chiyuv.
4) B'zman Hazeh
According to the Rambam and the Shagas Aryeh's understanding of the Rosh this halacha definately does not apply today. Even acc to the understanding in teh Rosh that it is m'din Yom Tov, it would not apply today since we have no para adumah.The Likuti Sichos however does say it applies today. Rav Shternbach in Moadim U'Zmanim also says there is a mitzvah to be tahor from tumas keri even though we are still tamei meis.
Halacha L'Ma'aseh B'zman Hazeh
The Sedei Chemed (mareches cheis klal 47) brings various poskim on this issue. Most poskim feel it is not applicable b'zman hazeh. The Be'er Heitev in Yorah Deia 373 brings poskim who hold this way. Also, in Even HaEzer siman 55, there is a halacha that a bas kohein is allowed to be m'tamei to her fiancee. The Beis Shmuel comments based on a Rashi in Yevamos that on Yom Tov she can't because she has to be tahor for the Regel. Rabbi Akiva Eiger and the Pischei Teshuva quote a Korbon Nesanel in Yoma that argues on this Beis Shmuel and says that this halacha does not apply today.
The Shoeil U'Meishiv wants to suggest that it applies today as a zecher l'mikdash. The Sefer Os Hi Meiolam is the only other poseik the Sedei Chemed brings who says it applies today because it is a din in kedushas Yom Tov.
Bottom line seems to be that most poskim would hold that the chiyuv to be m'taheir for the regel does not apply b'zman hazeh.