About 3 years ago I pushed myself until I did end up in the hospital. I had all the classic signs of a heart attack. I had a coworker take me to the ER where the Dr told me, luckily, I just had a stress induced panic attack, so I feel I have some understanding of this problem.
When is it appropriate to take sick leave/vacation time do to work stress / fear of burnout?
You should take it if you are thinking about how you feel stressed or that you are close to burning out.
The fact of the matter is no job is worth pushing yourself to illness, physical or mental.
You need to keep things in perspective and remember that the company you work for has no loyalty to you. Why would you push yourself to the point where you, physically or mentally collapse for them? More than likely if you end up reaching a point where you are unable to work for an extended time period due to illness they are going to replace you since you are no longer of an value to the company.
You must be proactive in preventing yourself from reaching a point of high stress / burnout.
How do you be proactive?
Keep track of your stress levels. Get a small pocket year calendar and every day after work write down a number from 1-10 of your stress level for the day, 1 being none to 10 being when you experience physical symptons due to stress. When the numbers start getting high and staying high that's when you need to schedule vacation time. Something simple as taking the next two Fridays off can do wonders for helping you relax. Spend those days relaxing, no house work, no stress. Sleep in, go out to lunch, catch a silly movie, etc.
Talk with your boss and keep him informed of your work situation and stress levels. Your boss should be open to having you communicate your feelings of your work environment. Most bosses like to feel useful and enjoy having employees come to them seeking advice, it inflates their ego. He might just end up being ok with pushing deadlines or getting you some help or finding other ways to lower your stress level.
Walk away and find a less stressful job. It might mean switching careers or taking a pay cut but in the end you'll be much more happy and healthy and that's all that should be important to you. Remember that an important job title and big paycheck mean nothing when you're dead.
Find an outlet to release the stress. Start a hobby, play video games, do a sport, meditate, join a gym or even get a weekly massage. Having a way to disconnect from work and do something you enjoy will work wonders for lowering stress levels. Make sure to make it a routine activity that you plan in advance to do. Saying you golf to relax doesn't do you any good if you don't actually get out there regularly and hit that ball.
Seek professional help. There is nothing wrong with seeking a trained therapist to help you learn to manage and deal with your stress. Not all therapists specialize in stress management so you may need to search around to find one. Also most health insurance plans do cover therapy costs so there should be no worry about the expense.