Is there a general study of the symmetries of tilings on surfaces? 
Conway, Goodman-Strauss & Burgiel classified them on $\mathbb S^2, \mathbb R^2$ and $\mathbb H^2$, with their 'Magic Theorem'. But I found nothing about general surfaces, except for the cylinder which is an easy consequence from the result above. 


I thought W. Thurston could have some results about this, but I found nothing. 



Classification of symmetries of tilings in surfaces?
As the OP points out, it is natural to study tilings from the viewpoint of symmetries acting on a space. From this perspecitve, finite subgroups of $SO(3)$ lead to tilings of $\mathbb{S}^2$ and the wallpaper groups yield tilings of $\mathbb{R}^2$.


Along the same lines, Chapter 7 of **Farb, Benson, and Dan Margalit. A Primer on Mapping Class Groups (PMS-49). Princeton University Press, 2011**, provides a very nice treatment of the problem for surfaces of genus $g\geq 2$ using the mapping class group. 


First, they discuss the $84(g − 1)$ theorem, aka [Hurwitz's automorphisms theorem](https://en.wikipedia.org/wiki/Hurwitz%27s_automorphisms_theorem), which shows that the orientation preserving symmetry group of $S\_g$ is at most $84(g-1)$. Using (orientable) covers of the orbifold quotient of the $(2,3,7)$ triangle group one can see this bound is sharp. 


Also, Chapter 13 of [Thurston's notes](http://library.msri.org/books/gt3m/) provides a good bit of background.


