Ticks tend to be bad here in NC and we used to get covered in them out on L.I. too (can you say tick research at Plum Island??)......may parents always used to get Vasiline on them and for the most part they would come right out on their own with no ill effects afterward. The last two ticks I had I pulled out with tweezers (one was in a really bad spot) and both times I had those spots itching for over a week. I am sure that pulling them out with tweezers will irritate the tick as much as anything else and have found that Vasiline is the lesser of two evils. If you are in doubt about whether you can safely remove ALL of the tick, go to the doc so they can get a magnifying lense over it and remove it properly for you.
The one tick I had in the spot I couldn't readily see only was found after I started feeling ill (flu like symptoms) and felt a weird twinge in that spot. I thought I had cut myself shaving or something. If not for the twinge, who knows how long he would have been there. I started to feel better after I pulled it out, but kept the thing in a ziplock bag in my freezer for a few weeks afterward just in case I had to go to the doc. I did identify it as one of the Lone Star ticks that carry Lyme disease. I was lucky that I didn't get sicker. Also, check out
this link for tick ID and symptoms of Lyme disease.