What is the correct place to use the word tumbleweed? Can we use it as a metaphor for a person who always irritates us?
Yes, tumbleweeds' traits can, and have been, applied to people:
"I'm just a tumbling tumbleweed!" - lyrics to a song written by Bob Nolan, an actor, poet and western music songwriter in the 1930's. It attributes the plant's characteristic trait of breaking off and rolling along the plains with the wind to a cowboy's lifestyle.
Edit: The question on whether an annoying person can be called a "tumbleweed" is more of a judgment call. It would depend on whether the person is annoying because they are shiftless and always on the move and at the whim of the "wind" or other external force, not because of their own motivation.
You can use pretty much any word metaphorically. However, tumbleweed is not particularly irritating, so using it to mean an irritating person is going to create confusion. Hearing it, many people are more likely to think you mean a person without a purpose, who goes in whatever direction the wind takes him.
From past usage, when a person is called (a) tumbleweed, it means that they are pretty useless or without function and not generally the irritating description being suggested. Tumbleweed has no real function, and therefore is seen as unnecessary.
I think this sense goes pretty well with general reference definition #2 in the comment section under the question.
It is used in such following situations:
A roaming party, where the sole intention is to roam, whilst partying. Thus giving the effect of a bush party that moves - Screw the bush party, let's tumbleweed.
Man im all out of tumbleweed you have a bag i can buy
Something to say during an uncomfortable silence or awkward pause in conversation. the conversation is so dead that a tumbleweed could be blowing through the people you are hanging out with like a desert - Silence "Tumbleweed..." Laughter
Tumbleweed moment where everyone fall silent not knowing how to react - tears or laughter.