Should Rep. Schiff refuse to comply with a Senate suboena to testify during an impeachment trial, that would be a completely unprecedented event in recent US history, so the results of such a refusal are impossible to predict with certainty.
First, it appear to me that Rep. Schiff would not be subpoenaed by the Senate no matter what. What would he testify to? Any role in the processing of the whistleblower's complaint? Rep. Schiff's testimony in such a situation would be irrelevant. Should there be documentation of his involvement in the process of handling the whistleblower's complaint, for example, Rep. Schiff's testimony wouldn't matter as any role he played would be documented. And if there's no documentation, there'd be no point in calling Rep. Schiff to testify. The only point in calling him to testify would be political, and in my opinion Senators won't want to poison House-Senate relations to score points on another member (because in the future, maybe they'd be the ones subject to such a subpoena, so let's not establish that precedent...).
And that applies to any situation I could see where Rep. Schiff's testimony could even be tangentially relevant to any impeachment trial. Rep. Schiff's testimony would be irrelevant because either events would be documented, or there would be no documentation and no reason to call him.
But, assuming we get to Rep. Schiff being called to testify in a Senate impeachment trial anyway...
It's a pretty strongly established aspect of US jurisprudence that the Senate and the House of Representatives have plenary power to set the rules for their own proceedings*, granted by the US Constitution, Article I, section 5:
Each House may determine the Rules of its Proceedings
In addition to that, the US Constitution's Article I, section 3 gives the Senate "sole power to try all impeachments":
The Senate shall have the sole Power to try all Impeachments.
Note that the only uses of the words "sole power" in the US Constitution are in granting the House of Representatives the power to impeach, and in granting the Senate the power to try all impeachments, making its use extremely significant (I'm not aware of any judicial ruling regarding interpretation of the impeachment clauses granting "sole power" so there's no precedent that I'm aware of, but I suspect no US court would want to involve itself in this.)
So the Senate should in theory be free to subpoena anyone they wanted to for impeachment trial testimony, and I can't see any way any court would block such a Senate subpoena given the "sole power" grant in the US Constitution.
So, Rep. Schiff would not be able to quash the subpoena.
What would happen should Rep. Schiff then defy the subpoena and refuse to testify?
As far as I can tell, that's a complete unknown. I can find no instances of anything similar happening in a Senate impeachment trial. The closest recent example of a US government official in any significant role that refused to testify before Congress is that of Attorney General Eric Holder, who refused to testify before the House, and was held in civil and criminal contempt of Congress in 2012.
Effectively nothing happened to Holder for his being held in criminal contempt of Congress as the Department of Justice refused to prosecute.
That would lead me to think Rep. Schiff would be able to effectively ignore the Senate subpoena.
However, Holder's case is significantly different from any hypothetical Schiff refusal to testify in two ways
- This would be an impeachment trial held by the entire US Senate, of much, much greater importance than a partisan-tinged investigation by a single House committee. The political and legal pressure on Rep. Schiff could very well be much, much stronger than what Holder faced and was able to withstand.
- Rep. Schiff is not in charge of the Department of Justice, so there would be no apparent "who wants to be the one to charge our boss with a crime" conflict of interest to perhaps protect him.
So, I don't think anyone can predict what would happen should Rep. Schiff attempt to defy a Senate subpoena to testify during any impeachment trial.
I'd speculate, however, that the results would highly depend on the importance of Rep. Schiff's testimony and how it might potentially relate to any facts established in the Senate trial. Should his testimony obviously be of marginal or no importance, I suspect Rep. Schiff would be able to ignore the subpoena with immunity.
In the entirely hypothetical situation where his testimony would be of utter, absolutely critical importance to the trial, and Rep. Schiff still refused to testify, it would not surprise me to see Rep. Schiff frog-marched live on TV onto the Senate floor live in irons for his testimony. I think that it's extremely, extremely unlikely for any situation to reach that point though, as Rep. Schiff would likely be able to delay such a possibility for a long enough time that his testimony would become moot no matter how critical it might have been earlier in the process.
Now, what would happen should we get to forced testimony, and Rep Schiff invoked his Constitutionally-protected right against self-incrimination? I suspect the Senate would be able to grant (or in this case, impose...) immunity to Rep. Schiff, removing his ability to incriminate himself and therefore his right to refuse to testify. But that would take time, too, increasing the chances his testimony would be moot. But maybe not.
So again, after a lot of back-and-forth we're all back to, "No one really knows what would happen."
It's just too unprecedented an event to be able to predict with certainty.
But do note that in all cases, "Nothing happened to Rep. Schiff for refusing to testify." is always, in my opinion, somewhere from completely certain to fairly likely.
* - and my Google skills are lacking to find actual cites for this.