The key to understanding what inserting zeros does is to understand two things: what samples represent in the time domain (because we want to insert zeros in the time domain), and what they represent in the frequency domain (because we want to know what it did to the spectrum).
First, sampling is a modulation (PCM—Pulse Code Modulation), equivalent to multiplying an impulse train by our analog signal, and creates images in the frequency domain. Here is an example spectrum represented by our samples; the original analog signal's spectrum is shown in green, and images in red:
These images are the price we pay to represent the analog signal as samples. As such, the usable bandwidth is from 0 Hz up to (but not including) half the sample rate. Above that is a backwards image of our original spectrum, with images repeated around multiples of the sample rate. It's OK, we remove the images when we convert back to analog, using a DAC's lowpass filter.
In the time domain, samples represent impulses. They are instantaneous values taken at a constant interval. Inserting zeros changes nothing except what we consider the sample rate.
For instance, sample a signal one time per second. That represents an impulse train, so consider playing it back raw, as an impulse train.
Now consider placing a zero-valued sample between each of the original samples. Consider playing it back as an impulse train but at twice the original rate, two times per second.
Can you see that nothing has changed in the signal, except the sample rate? Likewise, if we look at the spectrum, nothing has changed in the frequency domain—this is obvious, since the time domain signal has not changed.
However, our usable bandwidth has doubled. The first inverted image now lies in our usable band, now shown in green. It will not be removed by the DAC when played back through a DAC at the new, higher rate. And it will be a problem with any non-linear processing in the digital domain.
That is why we follow the zero-insertion with a lowpass filter (or combine the two steps for efficiency). Here it is again after proper filtering, below half the sample rate:
So, the answer is that nothing changes in proper integer sample rate conversion by zero-insertion, and the result is as perfect as the lowpass filter used to clean up the exposed images.