Do topological transitions only occur at Dirac points? Topological phase transitions happen when the band gap closes. It is not true that all band crossings are topological.
There are Dirac (linear) band crossings, quadratic band crossings, Dirac-like triply degenerate band crossings, double Dirac cone crossings, semi-Dirac transitions (linear in one direction and quadratic in another) etc. 
Even in 1D, all the band crossings I recall look linear. In 2D, all the band crossings I recall are Dirac cones. I feel like I have been told that some quadratic dispersions can be a topological transition but I am not sure if I remember correctly.
Are all topological phase transitions in electronic bands/ photonic bands linear/ Dirac points?
 A: I'll leave the aspect of classifying band closings at topological transitions to others, and focus on this statement:

Topological phase transitions happen when the band gap closes.

Although that's the standard story, there's a growing understanding that you can actually have topological transitions without gap closings. These so-called first-order topological transitions require some degree of interaction between electrons (or possibly other constituent particles). Topological transitions in non-interacting electron systems should still be continuous, and have an associated gap closing.
Relevant literature:


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*Amaricci et al. First order character and observable signatures of topological quantum phase transitions, Phys. Rev. Lett. 114, 185701 (2015).

*Imriška, Wang, and Troyer First order topological phase transition of the Haldane--Hubbard model, Phys. Rev. B 94, 035109 (2016)

*Juricic, Abergel and Balatsky First-order quantum phase transition in three-dimensional topological band insulators, Phys. Rev. B 95, 161403 (2017)

