IR-Receiver signal almost constant
A smaller resistor than 270 ohms will result in more current to your LED which will make it shine brighter... that might work a little bit, but don't go past what is spec'd in the datasheet for the LED. 

Honestly, I would probably buy a transimpedance amplifier from somewhere like texas advanced optoelectronic solutions (TAOS) as they're made specifically for this type of application... amplifying small light-->current signals often the result of biological phenomenon. The other thing you could try is a different wavelength of light. 

Red and IR LEDs are used in conjunction for a pulse oximeter but to detect heartbeat you only need one. You could try switching to a red LED to see if you get better results. I know from personal experience that positioning this type of sensor is extremely important. You need to make sure the detector is positioned right over an artery or else there will be no variation in light absorption when your heart beats.