Is it convenient to ask the journal's editor for two additional days to complete a review? I am assigned as a reviewer of an article for IEEE TNNLS (computer science), but unfortunately, I would be super busy during the week of the deadline. In fact, I need to submit a paper and also two postdoc applications during the same week. 

I'm already trying to do as much as I can before that week, but it'd make things much easier (with better quality) if I can postpone the review at least for two more days. So, I like to know if it looks unprofessional to ask the editor for that extension and whether it'd damage my review performance/reputation in that journal?
Getting a review in two days late for a journal article is pretty common. I wouldn't worry about anything less than being a week late. For conferences, deadlines can be tighter, but 2 days is nothing. Ideally you would let the editor know your review will be a couple of days late, but even if you didn't, it is not the end of the world.