Because of the conflicting evidence on the subject, many doctors recommend that their patients refrain from coloring their hair during pregnancy. Though we can’t give medical advice, we can share a little research on the subject that may convince you to hold off on coloring your hair until you consult with your health care provider.
Conflicting Evidence Abounds
Several studies have been conducted to assess the use of hair dye before and during pregnancy to measure the effect on the offspring that resulted in conflicting conclusions. For example, a 2005 study conducted by researchers from the Department of Epidemiology, School of Public Health, University of North Carolina, Chapel Hill, NC report possible risks associated with coloring your hair while pregnant. The researchers investigated the use of maternal hair dye use and its association with neuroblastoma in their offspring and found that use of hair dye in the month before and/or during pregnancy was associated with a moderately increased risk of neuroblastoma. 
Another 2005 study pooled data from childhood brain tumor cases and analyzed maternal beauty product exposure to determine if there was any correlation between the two. Though the data suggested that there was an increased risk of childhood brain tumors (CBT) associated with maternal exposure to beauty products, including hair dyes, the researchers concluded that the combined data from the study provided little evidence of an increased risk for CBT with mothers’ exposures to beauty products. 
Professionally Assessing All the Data
To date, the most recent research hails from a combined effort of researchers from across the country that was published in The Journal of Clinical and Aesthetic Dermatology in 2013. The doctors reviewed the epidemiological evidence that has been conducted on hair dye use and its association to certain cancers, including that from mother to fetus.
The authors noted that the research they reviewed is an attempt to calculate a risk and does not represent direct correlations. That being said, they go on to point out that “alarming” data does exist especially the link between hair-dye use in pregnancy and several childhood malignancies in offspring.
Because of this, they further advise that concerned pregnant women should avoid all hair coloring. 
The Experts Weigh In
When researching something as important as the risks of hair dye use during pregnancy, it is important that the leaders weigh in.
The Teratology Society is a respected source for cutting-edge research and authoritative information related to birth defects and other disorders of developmental origin. The society is composed of scientists from a variety of disciplines including clinicians, epidemiologists, public health professionals, and researchers . The society says this about coloring your hair during pregnancy:
“We know that only a small amount of any product applied to your scalp is actually absorbed into your system and therefore, little would be available to get to the developing baby. In addition, many women have dyed their hair during pregnancy with no known reports of negative outcomes. This information, in combination with the minimal absorption through the skin, makes hair treatment in pregnancy unlikely to be of concern.” 
(It should be noted that the fact sheet has a disclaimer that the information should not take the place of medical care or advice from your health care provider).
The American Cancer Institute takes the position that more research is necessary before hair dyes are conclusively linked with certain types of cancer. When it comes to pregnant women, they say the following:
“Not enough is known about hair dye use during pregnancy to know for sure if this is a problem, but doctors often recommend this just to be safe.” 
 McCall, EE. “Maternal Hair Dye Use and Risk of Neuroblastoma in Offspring.” Cancer Causes and Control 16.6 (2005): 743-48. Print.
 Efird ET, Holly EA, Cordier S, et al. Beauty product-related exposures and childhood brain tumors in seven countries: results from the SEARCH international brain tumor study. J Neuro-Onc. 2005;72:133–147.
 Saitta DO, Peter. “Is There a True Concern Regarding the Use of Hair Dye and Malignancy Development?” The Journal of Clinical and Aesthetic Dermatology 6.1 (2013): 39-46. Print.
 “Hair Treatments and Pregnancy.” Mother to Baby. Organization of Teratology Information Specialists. Web. 25 Nov. 2014. .
 “Hair Dyes.” American Cancer Institute. Web. 25 Nov. 2014. . http://www.cancer.gov/cancertopics/factsheet/Risk/hair-dyes