I'm sure that happens sometimes, but it's a good way to immediately get arrested for uttering those Forbidden Words in front of the rest of the jury, or to get trapped by the judge instructing you to affirm you will assess the case as a matter of fact under law and then getting arrested if you refuse to basically denounce your own beliefs.
Do you actually get arrested for that? I thought that they just asked you, and if you declared that you could not make a decision based on facts alone due to strongly-held moral beliefs, they simply dismissed you from consideration. I ask because of the precedent that tends to be set by the most common instance of this: religion. Jehovah's Witnesses, for instance, are pretty infamous for getting dismissed quickly, but say, being a Quaker or member of another pacifist faiths/denominations/secular movements on cases where the death penalty is in play tends to also be pretty common grounds for dismissal without arrest based on one's beliefs.
Of course, thinking on it, the issue you bring up is mentioning it in front of the rest of the jury; that sounds like the person in question got through the jury selection process without challenge and is now ambushing the entire court in an attempt to force a mistrial. If they're playing dirty pool with the rules, then I suppose getting arrested and/or fined is not as surprising.
Ah, there's also the possibility that the judge decides that the juror is simply spouting off in an attempt to get out of jury duty. In that case, being charged with contempt of court is also rather reasonable. That, however, isn't a strongly-held belief at all; it's just an opportunistic attempt to fake it in an attempt to skip one's civic duty. Sure as sin, if you bring it up, make sure you know what it is, make sure it's applicable to the trial in question, and absolutely, definitely don't suggest it as a reason you "recuse yourself" (because you can't; you can give them good reasons to dismiss you, but you cannot recuse yourself and you don't want it to look like that's the only reason you're doing it).