Is Murder in the Cathedral a "modern" drama?
I believe this play's themes can be considered "modern," and that the play itself, then, can be considered a modern drama.
If one looks at the play literally, there are still places in the world where it is clear that the church and state are at odds with one another, not just in terms of disparate views, but pertaining to aggressive politics. For example, China has strong laws in place to prevent any religious assembly that might cause a "disruption" to any part of society, especially with regard to restrictive Chinese political policies. In places like Iraq and Afghanistan, a religious group actually acts as the governing body in those countries—and many times one country may have several different religious sects vying for political control.
In a figurative sense, the play is modern as it shows two powers struggling for supremacy. In our country, there is a constant fight for power between the Republicans and the Democrats. Each side believes (as did Henry II and Becket) that they are in the right. Paranoia abounds, as is the case with Becket's priests. Temptation is rampant in politics, as seen with the four Tempters in the play.
On a more philosophical level, the play speaks to elements of the human experience: fear of death, temptation, hunger for power, and the ultimate sacrifice for what one believes.
If one is contemplating whether this play is modern, and whether it still speaks to its audience years after T.S. Eliot wrote it, the answer is yes.