Let $d=\deg(f)\ge2$. Since the height of $f^n(\alpha)$ grows like $C^{d^n}$ with $C>1$ (unless the orbit is finite), one would expect the sequence $f^n(\alpha)$ to contain only finitely many primes on probabilistic grounds. So people study other questions. For example, how large is the set of primes that divides at least one term in the sequence? See [1]. Or as joro indicates, one might look at the orbit behavior modulo $p$ for varying $p$; see for example [2] or [3]. Or one might ask how many terms in the sequence have a primitive prime divisor, that is, a prime dividing $f^n(\alpha)$ that does not divide $f^m(\alpha)$ for all $m<n$; see for example [4] and [5].

[1] Hamblenm Spencer and Jones, Rafe and Madhu, Kalyani, The density of primes in orbits of $z^d + c$, 2013, arXiv:1303.6513.

[2] Akbary, Amir and Ghioca, Dragos, Periods of orbits modulo primes, J. Number Theory 129 (2009), 2831-2842.

[3] Silverman, Joseph H., Variation of periods modulo $p$ in arithmetic dynamics, New York J. Math. 14 (2008), 601--616.

[4] Ingram, Patrick and Silverman, Joseph H., Primitive divisors in arithmetic dynamics, Math. Proc. Cambridge Philos. Soc. 146 (2009), 289-302.

[5] Gratton, Chad and Nguyen, Khoa and Tucker, Thomas J., ABC implies primitive prime divisors in arithmetic dynamics, 2012, arXiv:1208.2989.