The trees will shade the tomatoes to some extent and may cause problems with fruit maturity. Root competition from the trees can be very severe. Deciduous trees are especially adept at finding all the nutritious goodies you put into the soil around your tomatoes so the tomatoes wind up starving. Rules of thumb are to never plant within 30 feet of any large tree and preferably not within 50 feet. These may not be far enough given that a tree may have a root system that extends twice as far as the tree is high. If you have no choice but to grow near trees, consider growing in containers.
What about black walnut trees near my garden?
Allelopathy is the name for the effect of one plant suppressing growth of others. Best case example is the eastern Black Walnut. Juglone is a significant chemical component of the leaves, roots, and wood of the walnut. It is antagonistic to most of the nightshade family especially so to the tomato and potato. Under no conditions plant a tomato near where a black walnut is growing or has grown within the past 3 years. Above all, don't ever use black walnut leaves in the compost pile unless it has at least a year to age and break down the chemicals. Without the thorough breakdown, the juglone concentrates into the compost and acts as a systemic poison to tomatoes.
Juglone is produced by Eastern Black Walnut (Juglans Nigra), California black walnut (Juglans Hindsii), Butternut (Juglans Cinerea), Persian walnut aka english (Juglans Regia), japanese walnut (Juglans Ailantifolia), etc. In other words, by all members of the walnut family including about 20 others I did not name. The worst offenders by far are the black walnuts and then the butternut. Persian walnut is often grafted onto black walnut or paradox hybrid rootstock and the rootstock behaves much like a black walnut in producing large amounts of juglone. Persian walnut on persian rootstock does not produce enough juglone to significantly affect most plants growing in its rootzone, however, I know at least one person who tried growing tomatoes near a persian and found that some suppression of growth occurred.
The root system of a walnut tree may extend up to twice the height of the tree or more. The black walnut is tolerant to the juglone its roots and leaves produce. Dripline is irrelevant since the roots may extend far from the tree though juglone is more concentrated near the tree because of leaf accumulation and higher root density. Juglone is persistent in the soil for up to 3 years after a walnut tree is removed. Many people think the walnut trees in their yard are very valuable. This is usually not true because the tree may be small, have poor form, metal inclusions from fence, or other damage that reduces the value of the wood. Here are some current weblinks worth reviewing or use a search engine for “walnut juglone allelopathic”.
Does tomato gardening under trees hurt the trees?
Gardening disturbs the roots, gives unaccustomed nutrients, and watering
changes the natural cycle for the trees. Many trees live in a delicate
balance with soil fungi in which the trees leaves and roots provide sustenance
for the fungi and the fungi make nutrients available to the trees.
Gardening can destroy the fungal ecology and replace it with a microbe
dominated ecology which can kill the trees. This doesn’t mean it
will, just that there is the potential to do so. Again, container
gardening is much more attractive where trees are involved.