Would Theo face the death penalty post-Die Hard?

Potentially.

The other answer to this cites the SCOTUS ruling affirming the death penalty being applicable to this matter. However, there are several potential mitigating factors that could have prevented Theo from receiving the death penalty if the narrative of the film was to be believed.

They include:

1. Theo didn't kill anyone - While the law in California delineates felony murder (a capital offense) as including murder in the commission of a felony.However, the film clearly demonstrates that Theo did not commit any of the murders and he would have been unarmed when he was found by the police. A good defense attorney could have argued these points and worked a deal with the prosecution to limit Theo's sentence to life imprisonment, likely with out the possibility of parole.
2. Theo had information - The terrorists led by Hans Gruber certainly didn't finance their terror attack themselves. Theo was the key to discovering who did finance it, who provided the group with their false identification papers, who provide them with the military weaponry that they used and who stood to benefit from their mission if it was successful. THeo obviously had been in on the detailed planning of the attack and so he could easily parlay what he knew into a lesser sentence for himself.
3. Theo probably knew the "inside man" who tipped Gruber about the bonds in the vault Nakatomi Building - The film handwaves a very obvious fact: Gruber and his gang had explicitly detailed knowledge of building, the people who worked there, the security, the vault and its contents. They didn't guess what was in the building; they knew the exact amount and its exact location. The only way that would have been possible was with the help of an insider. The film never makes it clear who this person is (or was) and surprisingly for such an intelligent cop John McClane never seems to even consider this.

Theo could have parlayed this information (which would have been very valuable to the US government (who probably would have undertaken his prosecution due to the deaths of the two FBI agents and what were presumably FBI-employed or military pilots when the helicopter was brought down) into lighter sentence for himself. After all, everybody would have wanted to know who the inside man was and how far up the Nakatomi ladder the corruption went. THAT person would have then been eligible for the death penalty for being one the architects of the crime.

Yes, Theo would have eligible for the death penalty under 1988 California law, although it's likely that if he did go to trial it would have been as long as several years later due to the complexity of the case. However, he had so much to trade that it's very likely that he would have received a reduced penalty in exchange for information and his testimony at later court proceedings.

Yes. California has a Felony Murder law codified into law. The event of a homicide during the commission of a felony (Robbery or Mayhem on Theo's part), even if he did not commit the murder himself, would mean he can be charged with 1st Degree Murder. It was likely, and did happen, that murder/lethal force would be used during the crime, and Theo would or should have known this.

A 1987 SCOTUS ruling in Tison v. Arizona would have bolstered any such charge in the following year. Specifically:

Held: Although petitioners neither intended to kill the victims nor inflicted the fatal wounds, the record might support a finding that they had the culpable mental state of reckless indifference to human life. ... Because the Arizona Supreme Court affirmed these death sentences upon a finding that the defendants "intended, contemplated, or anticipated that lethal force would or might be used, or that life would or might be taken," the case must be remanded.

SCOTUS ruled that the use of Felony Murder rule on someone who did not do the killing themselves, due to the circumstances of the planned crime, can be valid.

The Tison brothers only escaped the death penalty afterwards because of their age at the time, having been under 20.