Might need a photo to know more of the scenario you're talking about here. I can't picture it in my mind.

Here is a link that might be useful: Virginia Beach Weather

If a tree grows back from a piece of the roots then it's a pretty invasive tree and you can just dig it out as it comes up around your newly planted tree. Its like how I have mexican petunias popping up in all my pots because the roots from last year survived the winter so all the mexican petunias are growing between the new plants I planted in the pots for the year (I didnt expect the mexican petunias to be hardy in New York City).
-Alex

Depending on what type of tree was broken, it will rot out sooner or later unless it resprouts. What will happen to a palm planted near it will also depend on the type of palm. But fallen trees are good for planting bromeliads on. And that looks even better with a few palms around it. You can do a lot with it and make it a feature in the garden. One very large tree I had fall down, Rain Tree - Albizia saman, started to resprout. I cut the top off leaving about 2 metres of trunk. I stood the trunk back up and now I'm going to plant some orchids on it. I'll cut off any new branches but the trunk (it's very hard wood) should last another 10 years. The thing is to use your imagination and a bit of research to make something of it.

Hi
First lets substitute the word "sometimes "for "often" when it comes to hurricanes?? particularly as we head into the main season?? lol
Would help a lot to know what kind of tree and what kind of palm.?? It's probably best to remove the old plants before replanting but you can get some fantastic shapes and contrasts .. I have a schizolobium that proudly displays marks from evey hurricane that passed through.lol
Where are you located ?? That would help a lot also?? gary

Ok lets say not too huge oak broke, very close to bottom, but root not come out and certified remover has not been called, owner just used axe to split visible remainings and planted hmm let say true date palm, very juvenile 1-2 year young over remainings of old oak. What happens to date palm will it be doing very defficient In next couple years and rise back or die right away?

I didn't realise you meant the only remains from the tree are all underground. As the tree rots away a hole will form there. If the palm is planted on top of the rotting tree it will end up being unstable and less likely to stay up at times of strong winds.

Exactly tree falls during strong winds and gets fixed by additional support of belts around trunk or wood sticks and ring as usual. What happens to planted palm in next years? Will it live?

Hi
having a hard time picturing this .Obviously the oak wasn't too large or you would,'t have been able to dig a hole. Oaks are VERY hardy
As tropic said if the oak dies you will be left with a hole if not it will push the palm out of the ground. Sounds like it's been pushed over recently??
you don't have another location .?? At least until you find out what happens to the oak?? gary

Oddly enough, I just did something very similar. There was a dead tree (I think it was Black Gum, but it was dead when I moved in) that came up through my deck. I had it cut down flat to the deck. Then I cut out the stump as best I could. It was about 1 foot below the deck, but still some tree stump below. I covered it with about a foot of cactus dirt mixed with sand and planted a Filibuster seedling. I am wondering if the stump will rot out, allowing the palm roots to dig in deep. I am blessed to live in the woods, so I don't get high winds.

Here is a link that might be useful: Virginia Beach Weather

If the hole formed by the rotting tree isn't too large then the palm roots might fill the hole. Just depends on the size. So without some sort of indication of dimensions it becomes a bit of a "How long is a piece of string" question.

It is about 3 feet across. There are some spots that is could find soil to the sides but a stump directly below. I think that palms usually grow roots straight down.

The only thing I can see is the palm falling over in the first high winds it encounters, much as tropicbreezent said. Since palms tend to depend on a deep taproot to keep themselves upright, the fact that there is a still the corpse of an old tree under it will prevent the palm from digging down deep enough to withstand the winds of a strong storm, much less a hurricane. While the in-ground root and trunk of the old tree will rot with time, it won't be quick.

When I moved into my house, there was a stump at one corner from an oak that had been removed a year prior. I broke it down further so that it was under the soil level. Now, three years after the oak was cut down, the stump is still there. Rotting, weak and falling apart on the outside, but still quite solid at the center. Even now with all that time to rot, I think planting a palm over it would make the palm very unstable because it just wouldn't be able to send a taproot deep enough. A seedling might be able to do it as it would send it's tiny taproot into any cracks in the stump and might even help to split it up as the seedling grows, but an already growing palm just wouldn't be able to handle it. At least I don't think so.