Is Einstein's mass energy equivalence ($E=mc^2$) relevant to biological systems?
Short answer

Any law of physics will of course be applicable to living beings. However, in the absence of mass gain and mass loss, the input = output equation for matter holds true and does not require any spontaneous creation and destruction of matter to equate. In other words, the effects of Einstein's discovery has no measurable/practical consequence on biological system (it is here but it is negligible) and the law of conservation of mass applies nicely.

Input and outputs of matter

There are two main sources of matter input:


Eating and drinking
inhaling (gas through the trachea and the lungs)


There are three main sources of matter output


excretion (urine and feces)
exhaling
sweating


The importance of inhaling and exhaling

Just to get an intuition about the importance of inhaling and exhaling matter in this input = output balance, note that when one make sport to lose weight, she/he principally lose weight through exhaling (see this popular video from SciHow). Similarly, trees mass mainly comes from  what they inhale (see this popular video form Veritasium)

Possible duplicate on Physics.SE

You might also want to read the How does mass leave the body when you lose weight? on Physics.SE.