Step 1: Understanding the Concept:
The objective is to identify the false statement regarding the tenets and customs of the Lingayats, a prominent Shaivite sect in South India.
Step 2: Detailed Explanation:
Statement (1): This statement is false. Lingayats are devout followers of Shiva. Their ultimate aspiration is the fusion of their soul (\textit{atma}) with the supreme entity, which they identify as Shiva. They do not venerate Vishnu, the principal deity of the Vaishnava tradition.
Statement (2): This statement is true. Lingayats do not practice cremation. They believe that upon death, a true devotee achieves unity with Shiva and therefore does not require the ritual of fire for bodily release. They observe ceremonial burial instead.
Statement (3): This statement is true. Lingayats worship Shiva in the form of the \textit{Ishtalinga}, a personal, portable linga symbolizing the divine essence.
Statement (4): This statement is true. This is the most distinguishing practice of the Lingayats, from which their name originates. Every member of the community wears an \textit{Ishtalinga}, typically in a small silver case, suspended by a thread over their shoulder.
Step 3: Final Answer:
The untrue statement asserts that the devotee will merge with Vishnu upon death. Lingayats are Shaivites, and their ultimate goal is unification with Shiva.