Health issues in multi-day running
Of all the parts of your body, your feet will probably be affected most on long and arduous races. Other areas to pay particular attention to are your back and stomach - and of course your mind!
Taping up might help prevent injury
Feet
Of all parts of your body, your feet will probably be affected most on long and arduous races. Keep on top of any blisters or sores by treating them as soon as you feel something, rather than waiting until later on.
After running several events, some runners to find that preventative taping helps. However the pattern to their taping various greatly from foot to foot and toe to toe, so it's a question of trial and error - you could try talking to a podiatrist before you go, though.
Back
If you don't have the back and shoulder strength to carry a rucksack, running with it for several days will lead to stiff and aching muscles. As you're unlikely to be able to take a sports masseur with you, I would advise in prevention rather than cure - get rucksack strong before you go.
As an aside, taking some oil and practising massage on your fellow competitors will gain you many friends (and earn you more than a life time of wages in gold!) though you'll probably be too knackered when you're there. If you do and you bump into me, I'd love one.
Hygiene
Take water purification tablets and use them not only for your drinking water but for when you clean out your cooking utensils (if the race affords you the luxury of such time). In warm desert races the number of competitors complaining of upset stomachs is high. Pay attention to your personal hygiene, too (and not for the benefit of those you're sharing a tent with).
Mind
Sweet treats for your mind
There will inevitably be times in a hard race where you question your sanity and what you are doing. A lot of runners over come these sections by debating with themselves how they might explain dropping out to people back at home. Others simply remind themselves of highs that the race will also give them.
Personally I got through the hardest moments of the Marathon des Sables by giving myself the luxury of one or two fruit pastilles. After several days of energy drinks, chewing sweets with a tasty fruity flavour was life-saving.