My heart rate was sometimes elevated when I was on high dosages of steroids due to my lack of sleep (b/c I totally became nocturnal). However, when I was down to 20mg my heart rate returned to normal with the exception of the day or two after I did a really intense workout. My body was steroid dependent for almost three years, so I learned how my body would react after doing a pretty hard workout.
I have to say that I think your trainer may be overworking your body, which in turn will cause your body to "burn out" without adequate sleep and rest. Your continuous elevated heart rate, even while resting, is your body letting you know it's exhausted! I think you should start out with one really hard workout a week with your trainer, and then eventually work your way up (with of course doing "normal" workouts 4-6 times a week. I don't think it is your prednisone that is making you want to pass out, I just think it's your body being worked to it's absolute max.
I am on Remicade and still get dizzy sometimes when I push myself to the max. But, I am now careful to listen to my body (even though I often don't want to) and rest when my heart rate is elevated and my body is telling me to do so. A good coach or trainer designs their workouts based on your previous workouts, rest, and of course, heart rate.
If I were you I would record your resting heart rate before you get out of bed. Once you establish your base, you'll be able to tell what your "new" normal is and know when your body is overworked and fatigued.
What I have learned is that a lot of personal trainers from big "chain" gyms (LA Fitness, etc) really aren't that smart or in tune to what is truly going on in one's body. Nor do they understand the extent or severity of an illness that one may have. Try and find a trainer who has a physical therapy background and has extensive knowledge of anatomy and how your body works (one with a degree is preferable). I have found that private personal training studios are the best. I have had personal training from both a "chain" gym and a private personal training studio and have to tell you, the quality and knowledge of trainer in the private studio was way better.
It's GREAT to push yourself to the max, but you definitely have to work your way up to it. Sometimes you get placed with "meat head" trainers that simply don't understand what is going on with you and your body, nor understand the severity of what your body has been through. It's so frustrating!
Definitely keep working out, but listen to your body. =) Try and get at least 8 hours of sleep a night, eat properly, and make sure you hydrate!