Can quantum entanglement cause nucleosynthesis to happen in stars? Even if there isn't enough heat and energy in the star to cause nucleosynthesis, could atoms quantum entangle to create a new atomic nuclei? Or would it be quantum tunneling - if this is possible?
You are referring to pyconuclear reactions - these are reactions that are initiated even when the temperature is effectively zero. They are caused by zeropoint oscillations of particles that are trapped in a deep potential well and hence are purely a quantum mechanical effect.

This is far from just of theoretical interest. It may be that pyconuclear reactions occur in the condensed matter of white dwarfs; possibly even initiating the carbon burning that can destroy the star in a type Ia supernova.