The Narasimhar sannidhi in Srirangam temple has Narasimha moorthy with a third eye like Shiva. Is the third eye of Lord Narasimha mentioned in any scriptures like Puranas or Upanishads?
Yes, There are references from Agamas that Lord Narasimha is three eyed. Many of them are not available for us online and the exact source is not known but excerpts or dhyana shlokas of Lord Narasimha are available in hard copies and are only available and given through disciplic succession. Some forms of Narasimha (like Yogananda Narasimha) are not very well discussed in the puranas but worshipped in temples and worshipped by upasakas.
One of the dhyanashlokas is as follows.
Quoting from Rickross's answer which quotes a collection of different agama procedures and shlokas of different gods. This is a book written by Krishnananda Agamvagisha. He is a scholar in Vedas and Tantra.
Nrisimha's body's complexion is like that of a mountain made of precious jewels. And that glow is terrifying the demons. His hands are resting on the thighs. He has three eyes and the body is decked with clothing made with Ratnas. His two hands are holding the Shankha and the Chakra. The face is terrifying because of the teeth and the tongue, which is coming out and which looks like a flame of fire. Half of his body is that of a lion and the other half is that of human. ....
This may not be the form which he saved prahlada but there are many other forms of him.
Hence, we can safely say that Lord Narasimha is three eyed.