While not a rigorous study, the Today Show investigated the claims made by "Your Baby Can Read," asking 10 different child development experts to weigh in. All ten experts had the same opinion:
Young children can be made to recognize or memorize words, but the brains of infants and toddlers are just not developed enough to actually learn to read at the level the way the enticing television ads claim they can.
The teaching method leaves a bit to be desired, as it requires the child to sit in front of the TV for the lessons. The American Academy of Pediatrics recommends against TV viewing for children under the age of 2, noting that:
Research on early brain development shows that babies and toddlers have a critical need for direct interactions with parents and other significant care givers (eg, child care providers) for healthy brain growth and the development of appropriate social, emotional, and cognitive skills. Therefore, exposing such young children to television programs should be discouraged.
There is evidence that children who learn to read earlier do better in school, but one has to wonder if this is an example of mistaking correlation for causation. More intelligent kids would be apt to learn to read sooner than their counterparts; these same children would do better in school.
Finally, bear in mind that there are no rigorous studies or findings published in scholarly journals that show that the teaching methods employed by "Your Baby Can Read" are effective.
On a personal note, I don't think using the "Your Baby Can Read" program will harm your child in anyway. Nor do I think it will have any long term impacts. However, I posit that you can get the same benefits — promoting early reading — by taking the time you would spend showing your child these videos and instead read to your child. You'd also save a chunk of change.