My asnwer follow Mr. Iangreen's logic, but I interpreted it differently.
" I do belive" is the ESL speakers' English. "I believe it" is the native speakers' English.
In Grammar, "I do believe" is possitive, and " I do not believe" is negative. Thus the word " Do" may goes either way, meanings: yes or no. However, "I belive it" is only possitve, you can not change it's meaning to negative by adding "not" to the statement. So, "I belive it" means possitive with no other alternative choice.
Let discuss Mr. Iangreen's example:
A: "I believe in God"
B: "I believe in God, too"
A: "Are you sure? You're just saying that...."
B: "I *do* believe in God! Honest!"
A: "Are you sure? You're just saying that...." and, B: "I *do* believe in God! Honest!". Here, A (native speaker) assumed that B's (ESL speaker) don't understand that " I believe" is a confirmation form of an agreement. B's responding was implying that he is also a native English Spearker, he understanding what he said. He reconfirmed, the choice between "do" and "do not", he will choose "do". In fact he chose "do" to reconfirm his statement. If the native English speaker said " I do belive" , it is more emphasis than " I belive". If the ESL speaker said " I do belive", it would be better to ask him/her " Are you sure?" It implies " do you understand ?".