DC Biasing a signal and what prevents the signal going to the power supply?
You are correct in your thinking that there is potential for interference from this type of arrangement. There are a few ways to solve it.



Figure 1. R6 and C4 provide some de-coupling for the supply of Q1. Source.

The circuit above isn't a great example but it shows the basic principle used in "the old days". Being powered from a battery there is a risk of interference from the output back onto the input due to the battery's internal resistance. R6 and C4 form a low-pass filter with a time constant of about 2 s and the voltage at the top of C4 will be held adequately stable for audio-frequency inputs.

The use of modern voltage regulators generally solves the problem altogether. In your example V+ would be held quite tightly by the voltage regulator. 


When your input goes negative which would tend to pull V+ down the voltage regulator would "open up" to pass more current to maintain the voltage.
When your input goes positive which would tend to push V+ up a bit the regulator will shut down a little to maintain or regulate the voltage.


The end result is that V+ can source or sink enough current to be unaffected by the injected current.