Step 1: Understanding the Concept:
The question asks to identify the incorrect statement about the beliefs and practices of the Lingayats, a distinct Shaivite tradition prominent in Southern India.
Step 2: Detailed Explanation:
Statement (1): This statement is incorrect. Lingayats are ardent devotees of Shiva. Their ultimate goal is the union of their soul (\textit{atma}) with the supreme reality, which they identify as Shiva. They do not worship Vishnu, who is the central deity of the Vaishnava tradition.
Statement (2): This statement is correct. Lingayats do not cremate their dead. They believe that on death, a true devotee is united with Shiva and therefore does not need the ritual of fire to be released from the body. They practice ceremonial burial instead.
Statement (3): This statement is correct. Lingayats worship Shiva in the form of the \textit{Ishtalinga}, a personal, portable linga which symbolizes the divine reality.
Statement (4): This statement is correct. This is the most characteristic practice of the Lingayats, from which their name is derived. Every member of the community wears an \textit{Ishtalinga}, usually in a small silver case, on a thread over their shoulder.
Step 3: Final Answer:
The statement that is not true is that the devotee will be united with Vishnu upon death. Lingayats are Shaivites, and their goal is union with Shiva.