In the autumn, when leaves are falling and losing their natural colors, do the trees still produce oxygen?

There is no oxygen production without photosynthesis, so when leaves die, they stop producing oxygen.

The world’s atmosphere does not run out of oxygen, however.
There are several mitigating factors.

1) The atmosphere has always had a huge surplus of oxygen, so winter reductions in oxygen production have only a tiny impact.
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2) Oxygen concentrations do vary in regions with sudden loss of leaves.

Tropical climates are not affected by autumn weather, and zones without any plant life (like the polar ice caps) don’t show much variation.

Global biomes such as the boreal forest and taiga show the most seasonal drop in atmospheric oxygen.

Naturally the winds stir the atmosphere and tend to equalize the concentrations, but there is still a measurable difference between summer and winter CO₂ and O₂.
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3) Even in deciduous plants, there may be some reduced winter photosynthesis from green pigments in stems, tree trunks, branches, twigs, buds, and so on.
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4) Even in areas where oxygen production is vastly reduced, oxygen consumption is also greatly reduced because dead or dormant plant life is not using up much for cellular respiration, and animal life is reduced because of seasonal death, dormancy, or migration.
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5) A great deal of oxygen production is from hot regions that do not undergo sudden deciduous leaf loss, and the plants there just continue to pump out oxygen as they continue to perform photosynthesis.
A good example is the tropical rain forests of the world — big contributors to the world’s supply of plankton.
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6) Ocean plankton is the greatest contributor to atmospheric oxygen.
Oceans contain uncounted billions of small photosynthetic organisms, which produce up to 80% of the world’s oxygen.

Here’s a http://berekely.edu site discussing the world’s biomes:
https://ucmp.berkeley.edu/exhibits/biomes/index.php