Mineral oil and petroleum products coat the hair and prevent oxygen from getting to it.
This can keep the hair from drying out in one way, but the effect that it could cause has been described as potentially "suffocating" the hair. Now of course hair is dead and can't actually breathe, but it needs some exposure to oxygen. If not this can make the hair even more dry and brittle. A poster once mentioned that mineral oil is used to put out small drum fires by welders. It does this by cutting off the oxygen supply.
Natural oils don't totally do this, and evaporate or wear away much more quickly.
In the winter, for me right now, the lesser of 2 evils is to use the Hollywood Beauty Castor Oil Hair Treatment on occasion to combat extreme dryness, especially for my ends. I only use a product with mineral oil in it if I really like it, it is far down on the ingredient list, and almost always it has to be a product that washes out, with the exception of the Hollywood Beauty.
More importantly, mineral oil and petroleum product can definitely clog the pores in your scalp. If you do use a pomade or "grease" or with these ingredients in it, do not get it anywhere near your scalp. Try to use it just on the ends. If you must oil your scalp, don't use these products to do it. Use a light, natural oil.
Some people here have no issues with mineral oil, petroleum or anything. Adrienne coats her ends in Vaseline with great results.
The best thing to do is research the info on it and figure out what is right for you.