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The planet only has three elements to work with: hydrogen, carbon, and oxygen. Organic life forms also exist on the planet. They have no access to other elements.
Given (based on) our organic technology today, how far can such a planet go with just three organic elements? What sort of technology may come about?
At the very least, you need CHNOPS -- Carbon, Hydrogen, Nitrogen, Oxygen, Phosphorus, and Sulfur in order to have organic life as we know it.
While Carbon, Hydrogen, Oxygen, and Nitrogen are found in nearly every area of organic chemistry, Phosphorus and Sulfur are far more highly concentrated in living creatures than in the environment. It is hypothesized that, due to Phosphorus and Sulfur being concentrated in living creatures, that these elements make a sort of "high water mark" for the amount of biomatter available in the entire ecosystems of Earth... That the only way to increase the total mass of living creatures at any given time, we would have to increase the amount of Phosphorus and Sulfur that is bioavailable.
So, the absolute minimum that a planet must have, before even discussing technology, is CHNOPS.
I noticed that the question has been expanded, then reverted in order to avoid invalidating answers made before adding Nitrogen... so I'm interpreting this as, the person asking wants to know what the effects of moving the various goalposts would be. As there is no way to naturally create a world without a wide range of elements, I'm assuming a very powerful engineer is involved in making this world.
First, the lack of iron (or copper) means a lack of a central cardiovascular system. No hemoglobin. No blood as we know it. (Well, there will be blood -- or more technically correct, there will be a fluid carrying nutrients around the body; it just won't carry oxygen.)
Our creatures will need to be very dependant on the amount of oxygen available in the air, as they'll be taking the air directly into their bodies through holes in their skin (or chitin), just as an insect does. This severely limits their size. It also greatly limits the metabolism, which means that all of the available energy will be used in the muscles and digestive organs, with any energy for cognition being reduced. Human brains need a LOT of oxygen.
Without anything to carry the oxygen in blood, we can't have technology, since we won't have large brains anywhere on the planet.
Let's move that goalpost a little further and add iron, giving us hemoglobin.
As an added bonus, this gives our planet a huge iron core. Hopefully it's still molten, so that our planet has a magnetic field, which will protect us from solar radiation and will help keep our atmosphere thick by keeping the solar wind at bay. We can use the metabolic energy that we'd otherwise be using from repairing damage from UV light, and instead use it to get our creatures' brains to grow even larger.
Except... neurons need Sodium. Specifically, they need sodium ion channels to pass electrochemical charges between neurons.
So far, with CHNOPS + Iron and Sodium, we have everything we need for basic tools... We have lignin, chitin, and the metabolism to grow large brains. With the lignin, we have wood and fiber, and with the chitin, we have a hard, easily shaped cutting tool.
We can use these to make all of the simple machines: Ramps, levers, and pulleys.
However, we don't have a way to work the iron into anything usable, since we can't melt it. I mean, we can try to melt it; we have enough carbon, oxygen, and hydrogen to make some very hot fires... but the only thing in our world that's solid at Iron's melting point of 1,538°c is carbon. And at this point, we don't have any way to keep the oxygen out of our carbon, so rather than melting iron into ingots, we just end up burning our entire smeltery down.
Time to move the goalposts again!
Let's add tungsten. It is the metal with the highest melting point. With an unusual amount of care and determination, we can build our smeltery out of tungsten.
So now we have iron and steel in tool form. We don't have stainless steel, so we're going to be constantly battling rust, that's something we can live with. We can plow fields easily, build railways, and have giant ships! The age of oil and steam has come! We can also have an industrial revolution.
We don't have glass, though. We do have diamonds, so we might be able to form some sort of lens. We might have telescopes and microscopes, but diamonds have a different refractive quality from glass.
We also don't have semiconductors. We do have electricity (since we have magnetite, which lets us bootstrap production of iron bar magnets), but no batteries, and electric transmission is extremely inefficient, since iron makes a terrible long-distance conductor.
So, the fastest communication that is reasonable is still the Pony Express or carrier pigeons. We could set up telegraph stations, but instead of miles between repeater stations, we'd have perhaps a thousand feet, and without batteries, each repeater station will need its own electric generator.
We can certainly create mechanical devices to perform calculations, like Babbage's Difference Engine... but that's our limit.
If we want an electric computer, we need copper, aluminum, and silicon.
At first, the silicon would go into the vacuum tubes, to allow the heat to escape more easily, but soon we'll discover the semiconducting qualities.
As an added bonus, with aluminum, we can now carry more than 10 people in our airplanes at a time... and we can have windows in our buildings as well, with the silicon!
Next goal post: Splitting the atom.
We'll need elements that are heavy and unstable enough to be split, though... but are stable enough that they don't deplete into chunks of lead by the time life evolves a brain on our planet. This means we need Uranium.
This also means that we can't escape having helium (due to alpha decay), and we can't escape having trace amounts of every element on the periodic table that exists between uranium and lead, to a few varying degrees.
There are still quite a few things left out, like lithium for high density batteries (when we get copper, we'll get basic batteries), mercury for efficient fluorescent bulbs, or neodymium for high strength permanent magnets, but these are refinements rather than foundations of technology.
If you're lucky, creatures on a planet with only CHNOPS, Iron, Sodium, Tungsten, Copper, Aluminum, Silicon, and Uranium might make it to the space age.
I doubt it, though.
Organic life as we know it needs also nitrogen and a bunch of other elements (no proteins without nitrogen, no chlorophyll without magnesium, no hemoglobin without iron, no ATP without phosphorus just to name a few). With just the 3 you list you won't go past sugars, alcohols and organic acids in term of complexity.
Your planet would be just a barren land.
With Hydrogen, Carbon, Nitrogen, and Oxygen?
You are missing the metals, which are at the core of any technological advancement that we have had as a species. Specifically Iron and Copper have been instrumental in the progress we have made. Similarly, you are also missing elements such as silicon, which forms the major component of landmass as we know it.
The temperature of outer space is roughly 3-4 Kelvin. With just the elements you have chosen, you will have a hard time having an atmosphere, as the elements may not have sufficient mass to form a strong gravitational pull. Depending on the distance from their sun, your planet will at best be sheets of ice for surface of varied thickness and liquid hydrogen etc flowing all over it - any gas would be lost from atmosphere into outer space. At worse, it will be a giant ball of gases constantly churning around a diffused centre, sublimating, condensing, vaporising, melting, solidifying and depositioning at different places within that ball.
Organic life as we know it on such a planet would be hard to come by because of the extreme physical conditions and lack of other elements needed for biological activity.
