Does replication continue till the end of life? What exactly do you mean? Does replication continue forever? If that's the case, the answer would be no. As a previous response said, short sequences located at the end of chromosomes shorten after each replication. The use of the telomerase is to prevent DNA from getting shorter after each replication, due to the inability of the replicating enzyme, the polymerase, to replicate the tips of a DNA strand. Telomerase solves this by filling in the unreplicated ends with parts of itself. As a result, the telomerase gets shorter and shorter after each replication. When the telomerase runs out, the DNA will begin to erod due to the fact that the tips of the strand are lost during ecah reolicationl. Eventually, the DNA will be lost and replication will cease to occur.
However, if your question was if cells kept replicating until death, then the answer is yes. This is because the fact that cells cannot replicate for eternity is one of the main reasons that we eventually die. In general, cell replication is what keeps us alive.
Then again, the rate of replication is also very important. Unlike most normal cells that are able to slow down the rates of replication when replication is not needed, cancer cells will keep replicating no matter what. Also, it is proven that some cancer cells have a mutation that allows the cells to produce telomeres, allowing the cells to replicate for eternity.
Lastly, the postition of DNA within the cells is in the nucleus, which is analogous to the core of a planet. The DNA is usually intricately packaged to form what people know as chromosomes.