The three white lines of Sri Lord Siva represent triune forces such as Creation-Preservation-Destruction, the Past-Present-Future, Begining-Middle-End -- like the Three Points in His and Sri Durga's Trident.
The Bindu -- represents the place where The Divine mother is asked to seat Herself on Her Lotus. The Bindu represents the Beggining point of manifestation of the Sri Chakra - the Universe
Vibhuti -- Sacred ashe, all things come from dust, the ashe is what is left over after fire burns an object, the building blocks of all creation are in that dust, hence primodial dust -- this dust emerge from the begining of the universe. Sri Lord Siva covers Himself in the dust of cementarys because He dissolves everything and then creates it again with this dust. Fire purifys and in Homas, Sacred Homa Fires are the Mouth Of God, so the ashe is the Prasad that God returns to us after our offerings are received by Him in Fire of Homas.
Sandalwoodpaste -- is said to have properties to withdraw disease from the body and to cool it when one has been doing intense meditations, anusthans, or tapas.
Kumkum - In the Lalita Sahasranama - In the Dhyana , It is the Divine Mother Herself who is the First ever to wear Kumkum- Vermillion powder on Her Ajna Chakra/ Thrid Eye. Kumkum, made from tumeric, is said to have many subtle qualites. It is in Her honor and Her Grace we are inspired to wear it.
The wearing of Vibhuti and Sandalwood Paste and Kumkum, in whatever Tilak you wear, protects the Ajna/Third Eye and invites the Deity to rests there. It also stimulates the Ajna Chakra/ Third Eye for spiritual awareness and meditation. Also, you are consecrating that all of your thoughts that you think or receive from others, your mind, your intellect, and spiritual subtle bodies to the Lotus Feet of God.
Divine Mother Herself, rests in Bindhu in the Kumkum and comes to protect Her Devotees immediately.
Wearing the Tilak is a form of worship and a tremendous auspicious rite to do very day.
There is no doubt about this.
It is highly suggested to wear the complete bindhu, tilak, or either one of the vibhuti, sandalwoodpaste, or kumkum
everyday.
I have recollected most of this from readings I found at Amritapuri Asham in Kerala, www.amritapuri.org, Sri Amritanandamayi; and from Sri Gayatri Vol 1 & 2 from www.karunamayi.org, Sri Karunamayi --
Incarnations of the Divine Mother.
In service of God, Guru, and the Lotus Children