Why is a 0 kt, vertical climb in a helicopter not safe near the ground?
A vertical take-off, known as a "towering" or "confined area" take-off, is normally only used when it is not possible to take-off and remain in ground effect before gaining speed and climbing away.  

For example, your operational need is to land in a clearing in a forest or perhaps a field surrounded by tall trees. In order to take off again, you need to  climb vertically until you've cleared the tops of the trees so that you can transition forward to gain speed.  You might need to use on at night (because you cannot see if there are any obstacles ahead) or when taking off from snow or sand when your vision will be obscured, making a vertical take-off safer.

It is technically defined as any take-off which requires you to climb vertically until out of ground effect.

In this answer, I've discussed the reasons why the power required to transition in ground effect to forward flight is much less than that required in a hover or a vertical climb out of ground effect.  Before attempting a towering take off, the pilot should conduct a power check to establish that they have enough power to hover OGE (out of ground effect).  If you do have enough power to hover OGE, then in theory, you could continue to climb vertically until the reduction in engine power output or lift (small effect) as you gain altitude reduces the power margin to the point where further climb is not possible without exceeding limits.

So, to the point of your question, why is it not safe?

The main reason is a scenario which is common to all helicopter operations, not just a towering take off.  Climbing vertically to 90 feet is exactly as dangerous as coming to a 90 foot hover in an arrival from forward flight.

This scenario is putting yourself into a shaded area of the height velocity diagram, known by pilots as the "avoid curve" or perhaps more accurately the   "dead man's" curve.  Importantly, this has nothing to do with how much power you have which of course, if the engine stops, suddenly becomes zero!



Source: Copters.com

In short, test pilots, using a fully loaded helicopter over level, prepared surfaces, try to enter auto-rotation from different combinations of height and speed.  At some point, they look at each other and say "a normal, well trained pilot, could only just auto-rotate safely from that combination of height and speed.  They plot these points on a graph, join the dots and say "see those shaded areas, don't go there".  It represents the areas where you will not have enough of the combination of potential and kinetic energy to make a safe landing.

As you can see, the towering take-off puts you deep inside the avoid curve.  This is the first safety concern.  If the engine stops, you will almost certainly crash.

I chose that curve, for an Enstrom F28, because it nicely illustrates that if you continued climbing vertically to 300 feet, you would pop out of the  top of the curve.  From this height, you could auto-rotate safely.

The second safety concern goes back to the power required. If you run out of power during the climb, the aircraft may start to descend into its own rotor vortex and can enter a vortex ring state.  In a confined area, you have no options for escaping it and certainly, all but the best pilots will not avoid a crash or, at best, a very heavy landing.  I'm a relatively low time private pilot and when I do my power checks, I want to see at least a 10% additional reserve over the measured power check.  There have been times when pilots have landed in confined areas but have then not had the power to hover OGE.  They have had to reduce the weight and/or wait for a lower temperature before trying again.

The third safety concern is the need for good skills and training.  When performing towering take-offs in typical sport and private helicopters, the pilot must be gentle on the controls. Relatively large or rapid control inputs cause large spikes in the power required. When you have plenty of speed, this doesn't really matter but when climbing vertically, you might be calling for more power than is available.  You must climb OGE slowly with gentle collective increases and, when clear of the top of the trees or other obstacle, ease the cyclic forward gently.  It is this last input which is especially risky.  If you push the cyclic forward as you would in a normal take off, you could find yourself running out of power and settling into the top of the tree with nothing left to give.  The safe way to do it is a slow climb but obviously, this leaves you in the shaded area for longer.  I was always relieved when I cleared the trees and got back into normal flight and the temptation, which must be resisted, is to increase your climb rate to reduce the time on the curve but of course, this reduces your power margins. The general principle is to use the power required and no more.  Gentle climb, gentle movements, be ready to pick up wind as you clear the obstacle, and you'd better have made sure before you get there that it's a head wind.

The fourth safety concern (and the last one to my knowledge) is not so immediately dangerous but can lead to trouble.  In a helicopter without a large power reserve, it is possible to reach a point where the power required exceeds the power available, especially if you catch a gust of wind. In this situation, the rotor RPMs will start to reduce or "droop". The only possible response is to lower the collective.  This normally means you have to give up and simply land and try again as above but any situation which leads to the rotor RPM dropping outside your control is to me, a safety concern.

All of this said, some pilots spend a lot of time operating in the shaded areas, for example in logging and power line check operations but they are appropriately trained, use machines chosen to reduce the risks, pay more insurance than the rest of us and accept the risk as "part of the job".