Since this is belief, not science, it is subjective, not objective, evidence itself would be subjective and open for interpretation. Therefore, your question is not applicable, other than in the way that I just answered it.
So, most importantly, would you be saying that some claims that we encounter can be be accepted on no objective evidence at all, while other claims require at least some objective evidence? If you answer nothing else in this response, I am most interested in this answer.
Also, of some importance, do you agree or disagree with the statement "Rationality is to be held above non-rationality as an approach to evaluating claims."?
Here's a question for you. Is it rational to believe in something that has not been proven false?
I think you mean to say "Is it rational to believe that something is false which has not been proven as such?", but you can correct me if I am wrong. I just want to be clear, so if I am taking you wrong it is not because I am trying to piss you off or anything, it is simply that I disagree with your wording and am failing to understand as a consequence. Mine may be wrong, instead, though, and it doesn't need to be a point of contention. If I am wrong, simply clarify it for me with a different wording, please.
So, putting it into a concrete example, what you are asking me is "If 'God Does Not Exist' is not proven, is it rational to believe that God does not exist?"
If I were to answer you by parroting the manner in which you answered me, I believe I would have to say "Since this is belief, not science, it is subjective, not objective, evidence itself would be subjective and open for interpretation. Therefore your question is not applicable, other than in the way I just answered it." Therefore, taking your own standard, believing that God Does Not Exist would be acceptable.
Indeed, believing that God Exists would seem to also be acceptable. One has to wonder something, though. If it is ok for Joe to believe God Exists, and for Peter to believe that God Does Not Exist, then is it ok for Mary to believe the completely incoherent combination of both "God Exists" and "God Does Not Exist"? Maybe it is psychologically impossible to believe two mutually exclusive claims at once, maybe not. But, if we suppose that it is possible, is it acceptable to believe both?
But, to answer you with my own approach:
As I have indicated elsewhere, No, it is not rational to believe something is false which has no objective evidence showing that it is false. This includes the claim "God Exists Is A False Claim" or, the equivalent claim "God Does Not Exist". But, I believe when I add that the claim "The Tooth Fairy Exists Is A False Claim" to the list of those claims for which there is no objective evidence (that there is no objective evidence that the Tooth Fairy doesn't exist), you will take offense on behalf of believers everywhere. Aside from the fact that you contend that there is evidence that the Tooth Fairy doesn't exist (a matter with which I contend), you seem to take offense to the mere comparison, rather than think that I am simply mistaken.
In any case, whatever the claim, let's choose the pantheon of God's as the list of unsupported claims, instead of thinking about the Tooth Fairy as one. This is in keeping with the OP, in any event.
Consider the following list of unproven claims (Not exhaustive, we're just looking at the first several of the Celtic gods, here. There are quite a number of additional Celtic gods and then all the other cultures to work through):
Macha does not exist.
Rhiannon does not exist.
Epona does not exist.
Aimed does not exist.
Blenus does not exist.
Borvo does not exist.
Grannus does not exist.
I think we agree that all of these claims are unproven. I am curious what your attitude toward each of these claims has been? In your life, have you acted as though "Borvo does not exist" is true, or have you acted as though "Borvo exists" is true. I contend that "neither" really isn't an option. People who accept that Borvo exists seek his help in gaining healing, while those who do not, refrain from seeking that help. Everyone has either sought that help from Borvo or not.
Indeed, I contend that when people discover that there are a pragmatically limitless number of these types of claims out there, they really don't start acting like each one has any need for evaluation for truth. They live their lives as though each and every one of the claims is true (that "X does not exist" is true), except for a very few of the claims.
I agree with you that it is not rational to hold that these claims are proven. But, do we really differ from each other in the following: It is not strictly rational to treat each of these, literally millions, of claims in a neutral fashion. Is it your contention that each of these claims is to be treated neutrally? As in, for example, should we wonder if we don't give homage to Borvo whether that will end in our sick Aunt dying of cancer?
If a compulsive patient came to you and said, "I am very worried about my Aunt. You see, she's ill and could easily die. Just to give her the best chance possible to survive, I am researching all the gods of healing that have been recorded, finding out how their aid was secured and following the historical ceremonies to the best of my ability. This is taking some time, as you can imagine! I have taken time off of work and moved back home with mother in order to make time for this."
You wouldn't want to knock him upside the head and say "ARE YOU CRAZY??" I know I certainly would, and I don't think I would be the irrational one in doing so.