Copyright laws on very old art works
3 attorney answers
The following is not legal advice and should not be relied on take or refrain from taking any action.
It depends on what you mean by "very old" and whether the work was published or not (i.e., made available to the general public).
Copyright protects both published and unpublished works but it does so for different terms. All unpublished works (pieces found in attics, archives, etc. and some that are in private collections) that were created before 1888 are in the public domain in the U.S. and may be freely reproduced. One odd wrinkle in our law: a work that was created before 1978 and first published between 1977 and 2003 is still protected by copyright until 2048 (no matter how old the work). That wrinkle may have been ironed out by now but I don't think so.
The artwork that you mention are all published works however. Each is in the public domain. In short, all works that were first published (anywhere in the world) before 1923 are now in the public domain. For a work first published after 1923 there is a complicated process to determine the duration of its copyright protection.
Now comes the interesting part: It is lawful to reproduce in whole or in part, take pictures of, and create derivative works from, artwork that is in the public domain. It is lawful to "copy the original." It is not lawful, however, to "copy a copy." So, while you are free to go to the Sistine Chapel and take as many pictures as they'll permit and then sell those pictures, you may not reproduce a picture that someone else has taken of the Sistine Chapel. Copyright attaches to that other person's picture and precludes you from reproducing it without permission of the owner of the copyright in the picture.
More interesting news: If the picture of the artwork that someone else took is of the entire work and has little if any of its own creativity then copyright does not protect that picture and it may be reproduced by anyone. But if the picture of the artwork that someone else took is of a particular angle of the art or is only a discrete part of the work or is lighted in a creative way or has any other indicia of creativity then copyright attaches to those creative elements of the picture and you may not reproduce them (or the picture) without permission.
If you're going to start a business by using the intellectual property created by others (either the old masters or those who photograph the old masters) then you need to hire an intellectual property to guide you through that process. I suggest that go to amazon.com or any other online book seller, query "public domain," and buy a few books on the subject. The have a chat with a copyright attorney to discuss the particular artwork that you want to sell on your tee-shirts.
You have received excellent answers to your question, but I would like to add one thing: some museums are very strict about photographs taken of their collections. You should check the particular museum’s policies regarding photos and commercial use of those photos before proceeding.
Ownership of the actual piece of art and the Exclusive Rights of Copyright are separate. The right you are considering is the right to reproduce the artwork. So first, the answer is NO, simply by owning the painting does not automatically grant the exclusive right to reproduce. But, the life of these exclusive rights are limited. The terms depend on when the art was created and published. There is a good measurement tool on the Copyright office (www.copyright.gov). But my initial analysis is that any copyright in art by the Masters would long since expired. The images of the terra cota warriors might be problematic because they would be quite recently taken. You should contact an attorney who works in this field. Best of luck.