Great comments by Yarrow -- some tips we men would know very little about!
First, sharing mascara with anyone--even a sister--is a setup for infection. Don't do it.
But the fact that you're OK until a little while after you put on your makeup probably means you don't have an infection--yet.
It's most likely either an allergy to the mascara or an allergy or irritation caused by putting liner INSIDE the eye. This technique is supposed to look dramatic, but if your eyes get red because of it, you're more likely to just look sick.
Eyes are designed to clear out junk that gets in there, and liner--to an eye--is the same as dust or flying crud. It's worse, of course, if you're allergic.
If you're still determined to try to use liner this way, check out the safer way to do it here:
http://www.ivillage.com/makeup-situation-lining-inside-eye/5-a-146608
Redness and burning can be caused by many things, including allergies, excessive dryness, even eyestrain. The best thing to do is to see an ophthalmologist for an exam.