Why can’t non-Mormons tour the inside?

The only people that can enter the Temple has to be worthy of a Temple recommend. Not all members are worthy of this. No unholy thing can enter the House of God. You have to follow and obey every commandment, attend all your meetings, have been Baptized by water and fire..that is being confirmed and receiving the gift of the Holy Ghost. Pay a full tithing, and love your neighbor as yourself. A non member can tour New Temples, or visit the Tabernacle and visitors centers. Which is where you can learn more about the why's..

Before each temple is dedicated, or church usually holds an open house. After being dedicated a remote is "Holiness to the Lord, the house of the Lord." The Lord asks us to grow to a certain level of spiritual maturity to worship in the temple, much like completing high school before entering college. If you are really interested in what temple are like and when happens there, you can go to temples.churchofjesuschrist.org. While it doesn't give every tiny detail, there is a video tour of the Rome temple, as well as information about the different services that are held there and when and where the next open houses will be held.