Please consider the conversation below:
Questioner: "Does Catholic Church teach that we will love everyone equally when we are in heaven?"
Answerer: "Since the Church teaches that God loves everyone equally, and that we will love 'like God' once we are in heaven, I would say that the answer is 'yes'."
Does the sentence "I would say that the answer is 'yes'" shown in above conversation have the same meaning as the sentence "I believe that the answer is 'yes'"?
The expressions are not necessarily interchangeable, but as written the sentences are essentially equivalent. There is always quibbling about connotations of words (at least, I like to quibble about them). But the structure of your answerer's sentence is that it presents argumentation which demonstrates why the answerer will provide the response he or she does, and then reveals the answer as a logical consequence of that argumentation.
Both of your sentences indicate that the answerer says yes to the question because he or she finds the argumentation compelling. He or she would say yes, based on that argumentation, and while the answerer might not personally believe that conclusion it would be odd to present this specific argumentation and conclusion as a direct answer to the question otherwise.
Because the sentences indicate the consequence of the arguments presented, in this case it would be a natural interpretation that the answerer both believes the answer is yes and would say the answer is yes because he or she believes the argumentation presented.
Your two sentences are not interchangeable and have different meanings.
The Questioner asks
Does Catholic Church teach that we will love everyone equally when we are in heaven?
The Answerer then gives a line of thinking about catholic doctrine and concludes with
I would say that the answer is 'yes'.
I would say so
it is a logical conclusion.
However, if the question was more simply
Will we love everyone equally when we are in heaven?
the simple answer
I believe so
may be appropriate since there is an open ended nature to the question, and it may involve certain opinions.
In casual language, both phrases do get used interchangeably, however "I would say so" has an amount of certainty about it (from the speaker's standpoint), whereas "I believe so" may have more, less, or equal certainty very much depending on the context involved.
