There is some truth behind this, according to Andrew Young, who cites numerous papers in his research:
Thermal damage (not really a “burn”) is possible under conditions of a
partial eclipse, when only a little of the Sun is exposed, and the
pupil opens up to adapt to the low overall light level; but it is
unlikely in normal daytime conditions.
He then goes on explaining how cases vary depending on age (younger people are more likely to damage their eyes than middle-aged people due to filtering) and other factors, such as drugs that dilate the pupils.
So, did Galileo go blind from looking at the Sun through his
telescope? The answer is, “Certainly not.”
Is it possible to injure your eyes by looking at the Sun? The answer
is “Yes, but you have to work at it under normal circumstances.”
Can you become totally blind from looking at the Sun with the naked
eye? The answer, according to Mulder, and from the cases of solar
retinitis in the literature, is “No.”
Is solar eye damage permanent? The answer seems to be, “Only if it is
severe.” About half the people with small scotomas recover from them
completely after some months or years.
I don't intend to minimize the seriousness of solar eye damage;
victims often lose the ability to read normal-sized print, for
example. But draconian pronouncements that “you should NEVER look at
the Sun” or assertions that you can become permanently and completely
blind are an over-reaction to the actual hazard. Sunsets can be viewed
safely, both with the naked eye and with binoculars, and most people
are already aware of this.
Basically, don't look at the Sun for too long (more than a few seconds) during an eclipse; in non-eclipse times, looking at the Sun should be done in moderation.