Part 1: Cell division in plants
No, all the cells do not divide all the time throughout their lives in plants. Growth by cell division occurs continuously throughout their life span in some specialised region known as meristems. This ensures the growth of plants throughout life. Some meristematic cells after differentiation get specialised in certain functions. The aggregation of such specialised cells to attain a common function are termed as permanent tissues. Differentiation cells usually lose their ability to undergo cell division until they are signalled to do so.
Part 2: Cell division in animals
No, all the cells do not divide all the time throughout their lives in animals. Hormonal influence regulates growth to a certain extent. Besides, most of the matured and differentiated cells like heart cells, neurons lose their ability of division. However, epidermal cells continue to divide. Hairs and nails show continuous growth indicating cell division taking place there.