If your roof is a Duro-Last roof, generally speaking you should not be using anything "black" on the roof. Petroleum based roofing products (which are usually black in color) are corrosive to the membrane and will damage the membrane over time.
Patches, repairs, TI's, etc. should be done using a hot-air welder and membrane material. In this process, hot air is used to literally melt the perimeter of a membrane patch over the hole/void, etc. No sealants or caulking would need to be used, especially petroleum- based ones.
However, this is not really a job for a newbie, especially as it involves patching very weathered membrane. The heat gun is not something you get at a Home Depot and the good ones cost about $500. Using a heat gun is an art in itself and takes some definite long-term practice to get proficient (we start our guys out by hand welding walkway - that way if they mess up it usually doesnt cause any leaks!). A patch that is correctly hot-air welded will not lift or separate as it is permanently fused to the existing membrane.
That said, since the roof is not warranted by Duro Last due to its age, you can do whatever you want. (If the roof is still warranted by Duro-Last, a Duro-Last approved contractor would need to perform the repairs to maintain the warranty) If you can get the black sealant off the membrane without damaging the surrounding areas, I would try and do so. Start by using water or simple green of something fairly gentle to clean things. You can work your way up through mineral spirits to MEK if needed, but be careful as the membrane is pretty old. Make sure you clean all the area where the new sealant will be applied. Sonneborn NP1, Sikaflex 1A or Vulkem 116 are all good quality sealants that will work to seal the patch. (Check you local roofers supply house for these)
BTW, congrats on getting 27 years out of your membrane roof. In my experiences Duro-Last seems to be one of the better single ply roof membranes on the market.