I remember being told that there are opinions that in the times of Moshiach, animal sacrifices will no longer be required. Where are the sources for this opinion?
It's based on a reading of the Rambam. In Moreh Nevuchim the Rambam devotes a large section of his sefer to make all mitzvos intelligible. He ascribes to the mitzvos associated with korbanos the need to give Jews a form of service that is akin to what they were used to in the world. Sort of like weaning them off from Avoda Zara. That being the case, korbanos would be unnecessary in the days of Moshiach (במהרה בימינו).
However, as many have pointed out, this cannot be said in the Rambam. The Rambam himself writes in Mishneh Torah that all Korbanos are chukim, and are beyond our understanding. He has great sections devoted to halachos of korbanos which will be relevant in the third and final Beis HaMikdash. Indeed, the Rambam davened (basically) the same tefillos we daven, and always expressed as we do the desire for the Beis HaMikdash and the opportunity to bring korbanos again.
Moreover, there are many p'sukim in Nach that explicitly tell us that there will be korbanos in the times of Moshiach. Examples: Yeshaya 56:7, Yirmiya 33:17-18, Zecharya 14:21, Yechezkel 43:22-25.