We know from the question Do Earth religions persist in Star Trek? that Gene Roddenberry wanted the world of Star Trek to be a world that had "outgrown" religion.
However, some Star Trek actors, such as Leonard Nimoy, were actively religious.
Have any of the franchise's cast members publicly disagreed with this theme?
It is unlikely that any of the well-known actors made such statements.
Your premise is not entirely correct. Roddenberry's thoughts are well known, but they didn't always translate to the screen. The very post you link to about Earth religions gives several examples of religion/spirituality throughout Trek (mostly from after his death). The producers and writers of Star Trek intentionally kept it secular, but secular doesn't always mean anti-religious. There is a good article about religion in Star Trek at Ex-Astris Scientia.
The strongest irreligious statements I recall are from Captain Picard in "Who Watches the Watchers":
PICARD: "Horrifying... Dr. Barron, your report describes how rational these people are. Millennia ago, they abandoned their belief in the supernatural. Now you are asking me to sabotage that achievement, to send them back into the dark ages of superstition and ignorance and fear? No!"
Patrick Stewart - an atheist and a member of the British Humanist Society - would probably agree with Picard to an extent. He believes in "positive vibes" but doesn't believe in organised religion.
I searched over a dozen actors' public statements to find opinions about religion in general, and cannot see that any of them would have had a problem with the show's philosophy. Most were either atheist, agnostic, or did not publicly speak about their faith at all.
William Shatner is a famous atheist, with many public statements such as:
Emotionally I would like to believe there is a life after death. Intellectually... I cannot accept the idea... as for myself, I have finally come to the conclusion that life is here and now... and nothing more.
https://www.celebatheists.com/wiki/William_Shatner
Michael Dorn is non-religious:
I haven't really affiliated myself with a specific religion throughout my life...
http://homegame.org/hgultra/Dorn.htm
Walter Koenig on religion:
...there have been more persecution and atrocities committed in the name of religion and religious freedom than anything else.
LeVar Burton was raised catholic but once said "the Roman Catholic Church is evil" and "religion has caused a great deal of harm in world history" (he later apologized).
Others were raised in traditionally religious homes and may have held on to their beliefs, but were never outspoken.
Leonard Nimoy is a known exception. He is well known for his faith, and based the Vulcan salute on a priestly tradition. Even he never expressed discontent that they didn't do more religious stories, or that Trek was somehow anti-religious. He felt that Judaistic traits were well represented in Trek, at least the non-supernatural ones.
If anything, it seems that the top-billed actors were basically all in agreement with Star Trek's message of enlightened, positive humanism without religious influence.
The closest thing to disagreement I could find is in Grace Lee Whitney's autobiography The Longest Trek. She played Yeomen Rand and was a convert to Judaism. In the chapter "The Great Bird of the Galaxy" she wrote about Roddenberry's personal problems with religion, and how that guided the show. That wasn't a complaint though. She identified with and understood where he was coming from.
In the episode, "Who Mourns For Adonis" Kirk gives the line, "Mankind has no need for gods. We find the one quite adequate."
I read somewhere that this was inserted at the insistence of the producers who wanted to maintain an Abrahamic religion at a minimum. One would also point to the many scenes in the chapel throughout TOS for essentially the same reason.
And then: (source)
"In the 1960s it may not have been opportune to create a decidedly atheist TV show, so Roddenberry may have been forced to drop (anti-)religious references to keep it at least secular."
