By a theorem of Eliashberg, two overtwisted contact structures on a 3-manifold which belong to the same homotopy class (as plane fields), are also isotopic (through contact structures). Is there an example known of two non-isotopic but homotopic tight contact structures on a contact 3-manifold?
Yes, there are known examples. See for example Corollary 1.3C of this paper of Eliashberg and Polterovich for an infinite sequence of examples on $T^3$. There is also a short paper of Akbulut and Matveyev giving examples on homology spheres. There may also be other examples known from the classification results of Giroux and Honda.
There are indeed many examples and one could cite a lot of papers but if you are interested in the history of this question then j.c.'s answer is not completely correct.
The first example is in Giroux's 94 paper Une structure de contact, même tendue, est plus ou moins tordue (which is explicitly cited by Eliashberg and Polterovich).