Just using a Y cable should not significantly affect the signal level to either amp as
@March Audio said above. The source impedance is usually low, in the 100-ohm to 1k-ohm range, and load impedance usually high, maybe 10k or more for a SS amp and 50k~100k for a tube amp.
For inputs it depends upon the internal circuit, but most inputs are left high-Z when off and can be safely plumbed (connected). Some discrete circuits (and potentially op-amp circuits but that would be unusual) may exhibit lower impedance and nonlinear behavior when driven whilst turned off, but again I expect that to be rare. (But I have had it happen to me in the past; YMMV.)
Edit: Outputs are a different beast and I would never Y two outputs together unless I knew it was OK or I had provided some sort of isolation between them (like a resistive summer circuit). The off one could load the on one, bad things can happen, etc.
