A lot of incorrect answers on this one.
The short answer is that as the law currently stands, mask requirements, generally speaking, do not infringe on civil liberties. People are of course free to argue that they know the true, secret meaning of the law, but they are arguing about whether mask requirements should be considered an infringement on civil liberties, not whether they actually are under the law as it actually exists.
The reason begins with understanding that contrary to popular belief, the law as it currently stands generally allows the government to tell you what to do and what not to do. The federal government does not have that broad power, but state governments do.
This is where answers like Phoog's and Dale M's go wrong. There is no general, freewheeling right to "liberty" or "to be left alone," at least not against the states. Governments generally have the power to regulate the vast majority of human interaction.
The Tenth Amendment takes basically the entire universe of power that a government could conceivably have and gives it to state governments. It makes exceptions for powers the Constitution gives the federal government (e.g., raising an army), and for powers the Constitution prohibits the states from exercising (coining money). Any powers outside of those groups are shared by the states and their people.
This powers conferred by the Tenth Amendment's delegation are known generally as "the police power," which are virtually boundless, as the law currently stands. Because it is so broad, it has been hard to define, but the definitive statement as to the scope of police power came from Commonwealth v. Alger, 61 Mass. 53, 85–86 (1851):
The power we allude to is rather the police power, the power vested in the legislature by the constitution, to make, ordain and establish all manner of wholesome and reasonable laws, statutes and ordinances, either with penalties or without, not repugnant to the constitution, as they shall judge to be for the good and welfare of the commonwealth, and of the subjects of the same.
It is much easier to perceive and realize the existence and sources of this power, than to mark its boundaries, or prescribe limits to its exercise. There are many cases in which such a power is exercised by all well ordered governments, and where its fitness is so obvious, that all well regulated minds will regard it as reasonable. Such are the laws to prohibit the use of warehouses for the storage of gunpowder near habitations or highways; to restrain the height to which wooden buildings may be erected in populous neighborhoods, and require them to be covered with slate or other incombustible material; to prohibit buildings from being used for hospitals for contagious diseases, or for the carrying on of noxious or offensive trades; to prohibit the raising of a dam, and causing stagnant water to spread over meadows, near inhabited villages, thereby raising noxious exhalations, injurious to health and dangerous to life.
The U.S. Supreme Court has repeatedly endorsed this formulation. See, e.g.,
Bos. Beer Co. v. State of Massachusetts, 97 U.S. 25, 27 (1877); Lake Shore & M. S. R. Co. v. State of Ohio, 173 U.S. 285, 297 (1899); Sweet v. Rechel, 159 U.S. 380, 398–99 (1895)
This leaves us with a situation where the law generally accepts that the states' police power generally grants them the authority to enact laws to protect the public health. This includes mask requirements, quarantines, and temperature checks.
Many people find that arrangement extreme or oppressive or unreasonable, but it remains the current state of the law. The power to prohibit you from entering a store without a mask comes from the same place as the power to prohibit you from poisoning your neighbor, sending your child to school without vaccinations, poking strangers with dirty hypodermic needles, or detonating a biological weapon. Do those laws infringe on civil liberties? Some people may think so, but as a legal matter, they are incorrect for the time being.
The best case to illustrate the point is Jacobson v. Commonwealth of Massachusetts, 197 U.S. 11 (1905), where Cambridge imposed a law generally requiring that everyone in the city be vaccinated against smallpox. When Jacobson refused, he was charged with violating the ordinance, put on trial and convicted. He challenged the vaccination requirement on the following bases:
his liberty is invaded when the state subjects him to fine or imprisonment for neglecting or refusing to submit to vaccination; that a compulsory vaccination law is unreasonable, arbitrary, and oppressive, and, therefore, hostile to the inherent right of every freeman to care for his own body and health in such way as to him seems best; and that the execution of such a law against one who objects to vaccination, no matter for what reason, is nothing short of an assault upon his person.
The Court rejected all of his arguments and held that the power to order vaccinations was squarely within the police power (again citing Alger):
The authority to determine for all what ought to be done in such an emergency must have been lodged somewhere or in some body; and surely it was appropriate for the legislature to refer that question, in the first instance, to a board of health composed of persons residing in the locality affected, and appointed, presumably, because of their fitness to determine such questions. To invest such a body with authority over such matters was not an unusual, nor an unreasonable or arbitrary, requirement. Upon the principle of self-defense, of paramount necessity, a community has the right to protect itself against an epidemic of disease which threatens the safety of its members.
(The whole case is worth a read. It's really surprising to see how much the current debates over COVID-19 sound just like the debates they were having back then.)
So to claim such a mask requirement infringes your civil liberties, you must be able to point to the liberty being infringed, and that's usually where the opponents of mask requirements fail. The argument is generally that the government can't tell people what to do, but, as discussed above, the government can tell you what to do. In fact, that's the point of governments: to tell people what to do (preferably with an eye toward maximizing everyone's freedom, health, and happiness).
Of course, there may be specific circumstances where a mask requirement infringes on a civil liberty. If you have a health condition that must be treated and that makes a mask dangerous to you, a mask requirement may infringe on your right to bodily integrity. This is why you'll generally see exemptions in state mask requirements for those who are medically unable to wear a mask.
But a religious challenge ("My religion prohibits wearing masks") is likely to fail, at least on First Amendment grounds, because "the right of free exercise does not relieve an individual of the obligation to comply with a 'valid and neutral law of general applicability on the ground that the law proscribes (or prescribes) conduct that his religion prescribes (or proscribes).'" Employment Div. v. Smith, 494 U.S. 872, 879 (1990). (The outcome may be different when you run the problem through state-specific religious-freedom laws.)
If people raise them in the comments, I'm happy to discuss how Ameican courts would analyze a challenge to a mask requirement in the context of a challenge based on any particular civil liberty.