Can a Non-Dvija wear the Tripundra Tilaka?

Question

Can a non-dvija wear the tripundra? (On the head only)

Answer 1

Only the twice born are to take this Tripundra on the forehead and the other parts of the body after carefully purifying the ashes by the mantra Agniriti Bhasma, etc. The Brâhmans, Ksattriyas, and Vais’yas are known as the twiceborn, (the Dvîjas). So the Dvîjas ought to take daily this Tripundra with great care. 
Srimad Devi Bhagavatham 11.15

However it may be understood that (from DB 11.15.1), Although Dvijas ought to have the mantravath dhaarana of tripundra, on forehead and other parts of body , it doesn't seem to forbid others from wearing tripundra, bhasma ,clarified in Devi Bhagavatham 11.13.(1-20)

ब्राह्मणैः क्षत्रियैर्वैश्यैः शूद्रेरपि च सङ्‌करैः । अपभ्रंशैर्धृतं भस्म त्रिपुण्ड्रोद्धूलनात्मना ॥ १३ ॥ This holding on of ashes is not prohibited to anybody; the Brâhmanas, Ksattriyas, Vais’yas, S’ûdras, mixed castes, and the vile classes all can use this Bhasma Tripundra
(Devi Bhagavatham 11.13.13)

They that hold Tripundras with devotion can have Bholâ Nâtha under their control; no distinction is made here between the Brâhmanas and Chândâlas
(Devi Bhagavatham 11.13.1-20)

PS- Alongside the other details , an anecdote in Srimad Devi Bhagavatham 11.15 says, when an iota of bhasma from Durvasa munis body fell on Kumbhipaka citizens, (when the sage happened to visit, look them downwards),they were freed from their sins, and had all pleasures. (Thus exalting the mahimaa of Bhasma)

Answer 2

While most sources are affirmative of the said issue, there are some sources who (rather regrettably) disagree.

For example Vṛddhahārītasmṛti states

रुद्रार्चनं त्रिपुण्ड्रस्य धारणं यत्र दृश्यते ।। तच्छूद्राणां विधिः प्रोक्तो न द्विजानां कदाचन । प्रतिलोमानुलोमानां दुर्गागणेशभैरवाः ।। पूजनीया यथार्हेण बिल्वचन्दनवारिणम् । यक्षराक्षसभूतानि विद्याधरगणस्तथा ।। चण्डालानामर्चनीया मद्यमांसनिषेवणाम् । स्ववर्णविहितं धर्ममेवं ज्ञात्वा समाचरेत् ।। रुद्रार्चनाद्ब्राह्मणस्तु शूद्रेण समतां व्रजेत् । यक्षभूतार्चनात् सद्यश्चण्डालत्वमवाप्नुयात् ।। न भस्म धारयेद्विप्रः परमापद्गतोऽपि वा । मोहाद्वै विधृताद्यस्तु ससुरापो भवेद्ध्रुवम् ।। तिर्यकपुण्ड्रधरं विप्रं पट्टाम्बरधरं तथा । श्वपाक इव वीक्षेत् न सम्भाषेत् कुत्रचित् । तस्मात् द्विजातिभिर्धार्यंमुर्द्धपुण्ड्रं विधानतः ।।

Translation: Where the worship of Rudra & wearing of the tripuṇḍra is seen, know it to be the rule for śūdras & never for the dvijas. Durgā, Gaṇeśa & Bhairava are to be worshipped by pratilomas (offspring of a marriage between a low-caste boy & a high-caste girl) & anulomas (offspring of a marriage between a high-caste boy & a low-caste girl) with (articles like) bilva-(leaves), sandalwood paste & water. Yakṣas, rākṣasas, ghosts & vidyādharas are to be worshipped by caṇḍālas by (articles like) wine & meat. Thus knowing the conduct according to own's own varṇa, one should act accordingly. A brāhmaṇa becomes on par with a śūdra on worshipping Rudra, on worshipping yakṣas & ghosts, he immediately becomes a caṇḍāla. Even in great distress a brāhmaṇa shouldn't wear bhasma, he who wears out of delusion surely becomes an alcohol-addict. One should look down upon a brāhmaṇa wearing a curved sign (i.e Śākta tilaka) & silken clothes to be on par with a caṇḍāla & shouldn't talk with him. Thus brāhmaṇas should diligently wear the urddhvapuṇḍra according to the rules. [Vṛddhahārītasmṛti:2:43.02-49]

Answer 3

Sri Skanda Purana 2.5.2.20-22.:

O Brāhmaṇa, a Brāhmaṇa should always have twelve Puṇḍras.[3] O son, Kṣatriyas should have four. It is prescribed (in Smṛtis) that Vaiśyas should have two Puṇḍras. It is laid down that women and Śūdras should have only one Puṇḍra.

The following are the parts of the body where a Brāhmaṇa should have his twelve sacred marks: forehead, belly, chest, hump of the neck, arms, ears, back, right side and lower part of the spine, and head, O sinless one.

A Kṣatriya should have the sacred marks on his forehead, chest and the arms; a Vaiśya on his forehead and chest and a Śūdra and women should wear the sacred mark on the forehead.

Why one should wear Bhasma and Tripundras.:

Srimad Devi Bhagwatam 11.12.:

1-20. Śrī Nārāyaṇa said :-- O Devarṣi Nārada! Hear now the great secret and the fruits of besmearing one’s body with ashes, yieding all desires. . .

.

The holding on of Bhaṣma (ashes) is according to the Vedas. Therefore one who does not apply it falls down. The Brāhmaṇas must use the Tripuṇḍras, repeating the mantra; and they are to besmear their whole body with ashes; if they don’t do so, they are surely fallen. He can never expect to get liberation even after koṭi births who does not besmear his body with ashes devotedly and who does not hold the Tripuṇḍras. O Nārada! The vile man who does not hold Bhaṣma duly, know the birth of that man as futile as is the birth of a hog. Consider that body as a burning ground which does not bear the Tripuṇḍra marks. The virtuous man should not cast a glance at him even. Fie on that forehead which does not carry the Tripuṇḍra! Fie on that village which has not a single temple of Śiva! Fie on that birth which is void of the worship of Śiva! Fie on that knowledge which is void of the knowledge of Śiva. Know them to be the slanderers of Śiva who mock at Tripuṇḍra. Those that put on the Tripuṇḍras, bear Śiva in their forehead. The Brāhmin who is Niragnik (without the holy fire) is not nice in every way. So if the worship of Śiva be not done with any Tripuṇḍra is not praiseworthy, even it be attended with abundance of other offerings. Those who do not besmear their bodies with ashes or who do not use the Tripuṇḍras, get their previous good deeds converted into bad ones.

21-42. Unless the Tripuṇḍra mark is taken up according to the Śāstras, the Vaidik Karmas (works) or those performed according to the Smritis prove injurious; the good works whatsoever done by any man count for nothing; the holy words heard seem as if unheard and the study of the Vedas counted as if not studied.

The study of the Vedas, Sacrifices, Charities, asceticisms, vows and fastings of that man, who does not use the Tripuṇḍra, all become fruitless. Without using Bhaṣma (ashes) if one wants liberation, then that desire is equivalent to live after taking poison. There is no doubt in this. The Creator has not made the forehead vertically high nor round; but he has made it slightly slanting and curved fit to have the Tripuṇḍra. Making thus the forehead, the Creator wants, as it were, to inform everyone that every one ought to use Tripuṇḍra marks; the curved lines also are made visible for this purpose. Still the igorant illiterate man does not put up the Tripuṇḍra. Unless the Brāhmaṇas use the curved Tripuṇḍras, their meditation won’t be successful; they will not have liberation, knowledge, nor their asceticism would bear any fruit. As the Śūdras have no right in the study of the Vedas, so the Brāhmaṇs have not any right to perform the worship of Śiva, etc., unless they use the Tripuṇḍras.

The merits of wearing it.:

Sri Skanda Purana 3.3.16.:

63b. He who applies Tripuṇḍraka redeems a thousand predecessors and a thousand yet to be born in his family.

64. In this world, he will enjoy all pleasures, a long life and freedom from illness and at the close of his life he will die peacefully.

65-69. He will then attain a splendid divine body possessing the eight valuable Aiśvaryas (such as Aṇimā etc.), will ride in a divine aerial chariot. He will be served by hundreds of celestial damsels. He will enjoy pleasures in different worlds in turns, in the regions of Vidyādharas, Siddhas, powerful and energetic Gandharvas, and the worlds of Indra and other Guardians of the world. He will enjoy in the cities of Prajāpatis and finally reach Brahmā’s region where he rejoices for a hundred Kalpas. He will then sport about in the region of Viṣṇu for the duration of the life of three hundred Brahmās (viz till the life span of Viṣṇu is over). Thereafter he attains Śiva’s (Rudra's) world and enjoys himself there for endless years (till Rudra's life span). Then he gets merged into Sadā Śiva. He is not born again.

• The Life Span of Indra and the Tridevas.

I hope this clarifies all your queries. Prd .