Any good exercise routine stresses your body to bring about an adaptation. So yes, in a sense weight training/HIIT/whatever can weaken your immune system in the short term by using resources (ie; protein, fats, calories, etc), altering your hormone balance (ie; elevating cortisol levels, etc), you get the picture. That is why if you are already ill it is a good idea to stay away from the training because you only risk weakening your immunity further and increasing your recovery time.
Weight training on it's own isn't going to boost your immune system, you need to sort your diet out, resting, hygiene, etc beforehand. If you're getting ill regularly make sure you're eating a decent diet, plenty of protein, fats, calories, fruit and veg, make sure you're getting plenty of rest with not too many late nights, make sure you take an early night if you're feeling run down rather than going out and getting smashed, make sure you're washing your hands regularly, etc.
Then think about integrating some weight training but at the same time you're going to have to revise your lifestyle further to account for the new exercise regime (ie; maybe a greater calorific intake, maybe more protein, maybe more sleep/rest per night, etc).