No, long hair has a appearance of falling out. But everyone's hair is constantly falling out. Hair generally falls out at a rate of 50 to 100 strands of hair a day. And regrowth of the hair generally matches that rate.
Longer hair is more noticeable than short. And our hairs are more likely to come out all at once when stressed. Long hair might also break releasing a portion of the strand, a problem that short hair doesn't have. When you are in the shower you aren't going to notice 2 or 3 short strands falling out. Strands of long hair are very noticeable.
Now, there is one related issued tied to this subject. Hair that is frequently pulled on will fall out faster. Hair that is kept in tight hair styles or is constantly being tugged on by it owner will fall out faster than normal. But that is due to the roots literally being tugged out of the head. The weight of hair alone can't do this. It is problem that favors long hair.