I could share my personal experience, such as the time when I told a boss of mine with whom I had a great relationship that I was going to take a day off before I got close to burning out, using the actual words "burning out", but even though all the anecdotal responses are heart warming and all,...

1) this sort of thing is almost completely situational; i.e. depends on your specific context

2) you won't know exactly how it will play out until you've said words that you can't take back

3) at which point, if you miscalculated, you employer now has info that puts you at a disadvantage

4) company-employee relationships by definition and structure give the company asymmetric advantages in issues such as information, financial negotiations, and time-line strategies.

I say all this, not to be a wet-blanket, but rather to offer an alternative. You can accomplish the same thing using different words. If you're burning out, don't ask for understanding from your company, do the right thing for yourself and do what is necessary to put yourself back on a sustainable path. If this includes giving a different reason for how you end up dealing with it, tell them what they need to hear, but, do not add 'worried about your job' to 'burnt out'.