1. To consume or require (some amount of) time. Often modified by phrases such as "a lot of," "a little," "too much," etc., between "up" and "time." We're going to have a seven-hour layover in Newark, so we'll need to find a way to take up time with the kids. The process isn't actually that difficult. The only real problem is that it takes up so much time.

2. To occupy someone's time, especially in a wasteful or irritating manner. In this usage, a possessive noun or pronoun is used between "up" and "time." My boss was always really unwelcoming whenever I came to her with a question, like I was taking up her time. Thank you for going over this with me, professor. I'm sorry for taking up your time.