Which animals are thought to possess some degree of a theory of mind?
Well, the areas most responsible for social cognition (or theory of mind) appear to be the right and left temporoparietal junctions, the right anterior superior temporal sulcus, the posterior cingulate, and the medial prefrontal cortex. [1][2] Of those, the right TPJ is perhaps the most important; for example, the response in the rTPJ shows a peak just at the time when someone’s thoughts are described. [3]

Given the perceived importance of the temporal lobe to theory of mind, it is probably a safe bet that you wouldn't find many non-mammals that experience ToM the way a human does.

Furthermore, among the many hypothesized 'signs' of ToM possession is the ability to recognize oneself in the mirror. This ability implies a level of self-awareness and identity awareness in the individual, and it appears that primates (chimpanzees and orangutans) are the only non-human animals that have successfully recognized themselves in the mirror. [4] Monkeys are also the only other animals to have confirmed rudimentary mirror neurons, which are hypothesized to play an important role in empathy and social cognition. However, there are pieces of evidence that suggest that mirror neuron networks may possibly exist in elephants, dogs, dolphins, and whales.

That being said, the evidence isn't wholly clear on this one, and is further augmented by the fact that we don't necessarily have a working cognitive model for ToM (outside of our verbal conceptual models, which are not exactly scientifically rigorous). Some researchers even believe that ToM is a uniquely human experience. However, until we can lock down what exactly it is we are looking for, this remains a tricky question to make any sort of scientifically rigorous assertion.



Sources used:

[1] Dodell-Feder, D., Koster-Hale, J., Bedny, M., Saxe, R. (2010). fMRI item analysis in a theory of mind task. NeuroImage. 

[2] Deen, B., Koldewyn, K., Kanwisher, N., Saxe, R., (2015) Functional Organization of Social Perception and Cognition in the Superior Temporal Sulcus. Cerebral Cortex

[3] Young, L., Cushman, R., Hauser, M., &amp; Saxe, R. (2007). The neural basis of the interaction between theory of mind and moral judgment. PNAS, 104(20), 8235-8240

[4] G.G. Gallup (1982). Self-awareness and the emergence of mind in primates.
Am. J. Primatol., pp. 237–248