You can indeed get sick this way, although it isn't all that dangerous.
Microflora is the term for bacteria that typically resides in any given part of the body. For instance, the skin has a whole range staphylococcus (including the infamous 'golden staph', which 10% of the population have in their nasal cavity). Various 'enterics' live in the lower gut. And as has already been pointed out, the same with some forms of yeast and fungi.
Now most of the time, transferring gut bacteria to the upper gut (stomach etc.) won't do a lot. But some enterics can cause problems, especially if your immune system is already compromised. So too with some viruses (the rotavirus, for instance, can be perpetuated this way, which is why it is often prevalent in infants). Not to mention many forms of parasitic worm.
But on the whole, there isn't a huge problem.
In fact many animals eat their own and other animal's faeces to maintain a steady balance of gut microflora, and this includes primates. So maybe instead of 'yakult', we should just be partaking in a steady passion for coprophagia instead!
Athon