I'm looking for a short story that I read in an anthology.
"Anachron" by Damon Knight; first published in If, January 1954, which is available at the Internet Archive.
In it, I think the theory was that you could time travel, but could never take anything that would be missed. As in, if you went to the past and took a lamp from a room and brought it back to the future, it would vanish as soon as someone would enter the room in the past so that there was no change to the timeline.
Right. They try to use the "time sphere" to steal a viola from the 17th century, but:
Peter looked down at the instrument in his hands. "I'd like to keep this, if I may."
"I'd be very happy to let you, but you can't."
As he spoke, the bubble went cloudy; the viola d'amore was gone like smoke.
"There, you see?"
"What sort of devil's trick is that?"
"It goes back . . . Later you'll see. I had that thing out once before, and this happened. When the sphere became transparent again, the viol was where I had found it."
"And your explanation for this?"
[. . .]
"Three centuries ago, then, at this particular time of day, someone was in that room. If the viola were gone, he or she would have noticed the fact. That would constitute an alteration of events already fixed; therefore it doesn't happen. For the same reason, I conjecture, we can't see into the sphere, or"—he probed at it with a fountain pen—"I thought not—or reach into it to touch anything; that would also constitute an alteration. And anything we put into the sphere while it is transparent comes out again when it becomes opaque. To put it very simply, we cannot alter the past."
So a guy figured out that he could take things from a far future where aliens/monsters/something seemed to have destroyed humanity and taken over,
The trouble was, of course, that Naples was green. Where the city ought to have been, a rankness had sprouted. Between the clumps of foliage he could catch occasional glimpses of gray-white that might equally well have been boulders or the wreckage of buildings. There was no movement. There was no shipping in the harbor.
[. . .]
One day further forward: now all the display cases had been looted; the museum, it would seem, was empty.
Given, that in five centuries the world, or at any rate the department of Campania, had been overrun by a race of Somethings, the human population being killed or driven out in the process; and that the conquerors take an interest in the museum's contents, which they have accordingly removed.
all the artifacts were in a warehouse, so they never got missed.
That's the museum basement, which the alien looters had overlooked:
Two hundred years, Peter guessed—A.D. 2700 to 2900 or thereabout—in which no one would enter the vault. Two hundred years of "unliquidated time".
He put the rheostat back to the beginning of that uninterrupted period. He drew out a small crate and prized it open.
Chessmen, ivory with gold inlay, Florentine, fourteenth century. Superb.
Another, from the opposite rack.
Tang figurines, horses and men, ten to fourteen inches high. Priceless.
In the ending, he flees into the machine as a fire destroys it in the present, and the last scene is him being stuck in the warehouse with the gateway closed as something big (presumably one of the aliens) enters the room.
Although the servants came with fire extinguishers and with buckets of water from the kitchen, and although the fire department was called, it was all quite useless. If five minutes the whole room was ablaze; in ten, as windows burst and walls buckled, the fire engulfed the second story. In twenty a mass of flaming timbers dropped into the vault through the hole Peter had made in the floor of the laboratory, utterly destroying the time-sphere apparatus and reaching shortly thereafter, as the authorities concerned were later to agree, an intensity of heat entirely sufficient to consume a human body without leaving any identifiable trace. For that reason alone, there was no trace of Peter's body to be found.
The sounds had just begun again when Peter saw the light from the time sphere turn ruddy and then wink out like a snuffed candle.
In the darkness, he heard the door open.