"And I'm chief engineer. So?"
Does Tuvok outrank her? The answer is "it's complicated". Tuvok is 3rd in line for the captaincy, while she's... quite a ways down that list. But if Chakotay or Janeway are on their feet, they have non-overlapping responsibilities. Tuvok can order her around if the ship is boarded: she can order him around if it'll help with a warp core breach.
Of course, none of that is really relevant, since the question at issue seems to be "Does B'ellanna have to scrape and grovel around Tuvok." The answer is no, regardless of rank. She'd have said the same to Chakotay or Janeway.
(Maybe not Janeway. But Tuvok certainly isn't the captain.)
He looks like he wants to hit her. That would be amazing, she hopes he does it. She's going to try and not throw the first punch here, though: she sent Leiutenant Carey to sickbay yesterday, and while that worked out okay in the end, probably Chakotay wouldn't like it if she did that again too soon.