I'm a new homeowner and purchased a house with basically NO insulation on the basement ceiling/livingspace floor. So far the project has gone well as I've used the faced batts to fill in the easy spots. However, I am running into quite a few spots where if I place the insulation up, it will be in direct contact with the copper heating pipes. I am hesitant to install the insulation in these spots as I am afraid this could cause a fire. Is this a posibility? Or am I just being overcareful for an amateur?

Paper won't ignite until it reaches approximately 450 degrees or more. Your copper heat pipes will never reach that temperature. not knowing where you are from, I will add that insulating the floor above will cool the space below. Be careful that you don't cause a pipe freezing situation somewhere. Be sure you are unstalling the batts with the paper facing towards the warm in winter space.

Thank you that was really helpful! I live in Eastern New York. So far the temperature has not reached below 45 degrees in the basement on the coldest days we have had this winter(this was before i put in any new insulation). It's about 80% complete now and staying at around 48 to 50 degrees. Do you think it's at risk for freezing once the insulation is completed given that information?

There is nothing to worry about covering or encapsulating the pipes with the insulation.
The only thing you need to remember is
Keep the vapor barrier. That is the side with the paper or foil towards the heated side of the house. In otherwords you should be seeing the insulation not the vapor barrier once its up.
If both the basement and the first floor are heated and you are not heating the basement or only doing it once and a while. Then there should be NO vapor barrier at all. It will trap condensation and that is not good.
If the house is old. around 75 or more. Be sure that your wiring is not knob and tube. If it is you do not want to cover that as it becomes a fire hazard.
Also be real carefull when putting the insulation over the boiler or any other heating equipment. If it falls down it can also cause fire safety issues. It should be covered with 5/8 sheetrock and screwed tight to the ceiling in those areas.

the most important are is around the edge of the basement above the basement walls where the framing sets on that area. A lot of heat loss goes out that area.