Cornell psychologist Edward Titchener (who studied under Wundt) tested the ability to detect unseen staring and published in Science (1898). His conclusion: people cannot detect when they are being stared at, though a great many believe they can.
Though others have tested this `ability' over the years (e.g., Rupert Sheldrake), the results are mixed, and when statistically significant results are got, this is usually because of sloppy procedure.
Titchener would be rolling in his grave to know this sh*t is still being discussed.
Here's a short summary (source):
[Titchener] reported that over the years he had conducted a large number of informal tests and found no evidence for this particular claim. As far as Titchener was concerned people were not able to demonstrate their widely held belief.
He went on to provide a very good normal explanation for why people have this belief. First he noted that humans have forward facing vision, which leaves us exposed to the rear and he suggested that when in a situation where you are forced to present your back to a group of people, that there may be some psychological discomfort in that. He went as far to say that our ancestors must surely have devoted constant care to the defence of their backs. Titchener claimed that this back vigilance is the first element of staring detection: that people protect their backs by being aware of the environment behind them.
Once the feeling to turn around has formed, it is followed by an executing of the behavioural component: turning the head around and examining the back environment. Attention moves across the back of the room, scanning it to update their information as to what is actually going on.
Titchener next turned his attention to what might be happening behind the individual. He noted that his students could be engaged in a range of different behaviours (playing with their hair, eating food etc) and that it should be expected that some of the people sitting behind may be staring in the general direction of the individual. When they turn around, they disturb the visual field for those people who happen to have been looking in their general direction. This movement is a strong stimulus for people sitting behind, which they are required to attend to. There may in fact be any number of people who have the person in their visual field and who suddenly respond to the movement of the person looking around.
It is this coincidence which Titchener argued is the basis for the belief. Why do people feel the special tingling in their neck though? Titchener remarked that this no different from the feeling experienced in the bottom after sitting down for a long time!