The following is pasted from wikipedia page; (as heard from acharyas, the information below is indeed correct).
Here, the 9 prescribed kinds of activities in Bhakthi are listed. Having tatoo is not one of the nine prescribed activities of devotion.
“Bhagavata Purana and Navaratnamalika
The Navaratnamalika (garland of nine gems), nine forms of bhakti are listed: (1) śravaṇa (listening to ancient texts), (2) kīrtana (praying), (3) smaraṇa (remembering teachings in ancient texts), (4) pāda-sevana (service to the feet), (5) archana (worshiping), (6) namaskar or vandana (bowing to the divine), (7) dāsya (service to the divine), (8) sākhyatva (friendship with the divine), and (9) ātma-nivedana (self-surrender to the divine).
The Bhagavata Purana teaches nine similar facets of bhakti. “
Furthermore, a Saranagatha has declared that one very self is property of Perumal. The body one resides in is also Perumal’s property. We do not have any right to tatoo the body or torture it in other such ways.
For those who are not realised their saranagatha position yet, there are some varnashrama based regulations / restrictions regarding whether or not one can tatoo the body.