A non-memory foam mattress can be a healthier alternative to synthetic, petroleum-based foam. Most memory foam mattresses on the market today are perfectly safe. They must adhere to strict manufacturing standards that forbid them from containing potentially toxic materials.
Even so, memory foam is not as good as some other materials at fighting allergens. It is also not as healthy for the environment as natural materials. Some people may also have sensitivities to some of the chemicals inside memory foam mattresses.
So memory foam is safe, but it is not necessarily the healthiest choice. Healthier alternatives include latex foam mattresses, latex hybrid mattresses, and natural materials like wool and kapok.