In accordance to my limited knowledge of biology, asexual organisms exactly copy their DNA as there is no "contaminant", which in sexual reproduction is the other organism's DNA.
It is not necessary that no mutation occurs in the system where no gene recombination or "contaminant" takes place. There are mutations (which are sudden abrupt inheritable changes)
There can be errors in genome which can cause mutations.
Take example of replication of DNA-:
There are many enzymes linked with it : DNA polymerases ( I, II,III) in prokaryotes where asexual reproduction occurs which add nucleotides to make replicas of DNA of parent ( you can check the complete process at wikipedia)
Come to the point : DNA polymerase III, after synthesising dna chains has ability of proof reading. It can correct the mismatches ( hope you know that adenine bonds with thymine and cytosine with guanine, they are nitrogen bases of dna)
DNA polymerase III can correct the mismatches but it is not error proof- it can't distinguish between thymine and uracil (another nitrogen base ; thymine is methylated uracil)
This is only one possible error, there can be many such hence asexual reproduction isn't error proof