I believe the following sentence is grammatically correct:
I do like burgers.
Meaning that the speaker wants to emphasize that he likes burgers.
What about something like:
I do am happy.
She does is married.
when the speaker wants to emphasize their current situation? Is this grammatically correct? If not, are there any other alternatives to emphasize while using the verb to be?
No.
The kind of emphasis you are talking about is called verum focus because it emphasizes the truth of the sentence being uttered.
In sentences with an auxiliary verb (be, have, or a modal verb like might, can, must, etc.) it is that auxiliary verb that gets focused.
The verb do is only used when there is no auxiliary verb in the sentence. This is the same do that shows up in questions and with negation.
So just like you can't say:
You also can't say:
For sentences with auxiliary verbs, just focus the first auxiliary.
"I am indeed happy", "She is indeed married". you can move "indeed" around: "Indeed I am happy", "I am happy indeed". The last has added emphasis on "happy" to my ear; it's like "I am happy, very happy" or "I am _definitely_happy!"
You can say "I, indeed, am happy", but it's a little more unusual. It put the emphasis on "i".
