We now know a lot more about the potential health risks associated with deteriorating lead paint, with those most at risk being infants. If your home is older than 40 years, it might well have lead paint.
Another building material used in homes during the 1960s and 70s is aluminum wiring. This started to be used as an alternative to copper wiring but was found to be defective over time. If your property has this type of wiring, it will need to be replaced.
One material that we now know to be very hazardous is asbestos. However, during the 1930s and 50s asbestos insulation was frequently used. This will need to be taken care of professionally if you find your home has it.
Radon is something that cannot be discovered by sight or smell but your older property could have higher than safe levels of it. A test should be done to determine radon levels in the home.
You are more likely to find these types of materials in homes that have had the same owner for many years. Keep in mind also that most of these materials will need to be dealt with by a licensed professional and this could take up a good chunk of your remodeling budget.