Low voltage transformer on both doorbell & furnace
I doubt you've overloaded a 40 VA transformer. 

I think what's happened is the Ring is designed to power itself by leaking a small amount of current through the doorbell ringer (but not enough for the ringer to actually ring).  Similar to how old-school dimmers leak current through the incandescent bulb.  However in your case, your ringer is too small/high-impedance, and it can't leak enough power to stay alive. 

The answer is put something in parallel with the doorbell ringer so current can leak around it.  

You could try fitting a Lutron LUT-MLC device there (in parallel with the doorbell ringer) and see if that clears the problem.  The LUT-MLC allows additional leakage to make inline-powered devices happy.