Can the direct current output of a photodiode pass through a capacitor? If so what effect are they having in this circuit?
Can DC go thru a Capacitor?  NO
Is it DC that the Light Sensor picks up from the pulsing LED going thru a finger?  NO
What might it look like?
Time domain and Freq. Domain with 0 to 100 Hz linear bins.

This is my hand-drawn image of a heartbeat at 120 BPM with log dB vs linear Fourier spectrum to 100Hz.
What is the circuit?
First Caps reduce the voltage in half but also act as passive LPF followed by a Sallen-Key BandPass Filter (BPF) . The upward slope amplifies the downward slope of the spectrum of the heartbeat.
You can say it is basically a band-limited active differentiator.  The heart signal is differentiated and higher f noise is attenuated. 
The output would amplify the middle pulse of the QRST wave.