I came to Dubai on December 25, 2016, on a tourist visa good for 90 days. How long can I stay. My ticket shows that my return to the Philippines is on February 20. Is that the date I have to leave? Should I leave on the date on my ticket or can I stay longer?
Typically this type of visa is valid for sixty days from the date of issue and the maximum duration of each stay is 90 days. Thus legally you must leave before 90 days after you entered, and you must enter before sixty days after the visa issue date else the visa becomes void. See
90-Day Long-Term Visit Visa (Individuals)
Valid for 60 days from the issue date and non-renewable. Valid for 90 days from the arrival date and non-renewable
Within those limits you can leave whenever you want. You may choose to leave on the return day on your ticket, or extend you stay by paying a fee to your airline (if allowed) or even change your departure to leave earlier. That's between you and the airline. In your case the absolute deadline when you must leave is 24th March.
Bear in mind however that subsequent visa applications might be viewed unfavorable if on your previous visit you stayed significantly longer than the period you asked for in your visa application.