As discussed in the previous mitzvah, one may not use consecrated animals in any way. That mitzvah addressed working with such animals; in this mitzvah, we are told that they may not be sheared for their wool.
The reason is as we said in Mitzvah #483: we must keep far away from making personal use of things that have been dedicated to the Temple. We must have the utmost respect for the Temple service and it’s a metaphorical “slap in the face” to God for us to make use of things that have been sanctified to Him.
This mitzvah applies in all times and places. It is discussed in the Talmud in tractate Bechoros on pages 24b-26b and codified in the Mishneh Torah in the first chapter of Hilchos Me’ilah. This mitzvah is #114 of the 365 negative mitzvos in the Rambam’s Sefer HaMitzvos; it is not listed in the Sefer HaMitzvos HaKatzar of the Chofetz Chaim.