Equation of motion for photon $$ \Sigma \frac{dt}{d\lambda} = aL\left(1-\frac{r^2+a^2}{\Delta}\right) + \omega\left(\frac{\left(r^2+a^2\right)^2}{\Delta}-a^2 \sin ^2\theta\right)\ , $$ $$ \Sigma\frac{dr}{d\lambda} = \sqrt{R(r)}=\sqrt{\left( \omega\left(r^2+a^2\right)-aL\right)^2-K\Delta}\ , $$ $$ \Sigma\frac{d\theta}{d\lambda} = \sqrt{\Theta(\theta)} = \sqrt{K-\left(\frac{L}{\sin\theta}-a\omega\sin\theta\right)^2}\ , $$ $$ \Sigma\frac{d\phi}{d\lambda} = L\left(\frac{1}{\sin^2\theta}- \frac{a^2}{\Delta}\right)+a\omega \left(\frac{r^2+a^2}{\Delta}-1\right)\ , $$ where $K$ is the Carter constant of the motion.
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I wrote some open-source Python code for that purpose: https://github.com/bcrowell/karl . I believe the code used in the actual movie is not open source. There is a very complete discussion of the techniques by Riazuelo: https://arxiv.org/abs/1511.06025 . I don't think Riazuelo's source code is available. Mine only currently handles Schwarzschild black holes, i.e., not rotating ones like in the movie.
My video simulation of falling into a black hole: https://youtu.be/z-H-PipYCKc
This lecture https://www.youtube.com/watch?v=lCWwsg6CtL0 by Riazuelo (in French) has some nice videos.
A related question, with an answer from me that includes some still images and discussion of the physics: What will the universe look like for anyone falling into a black hole?
I hadn't known until seeing Void's comment today that there were two open-source projects for this by Riccardo Antonelli: starless and schwarzschild. Wow, nice!