If different but in tact molecules of DNA were extracted from my body, for example hair sample, blood sample, and mouth swab, how much variation would there be between DNA molecules?
The phenomenon is called mosaic single-nucleotide variations (SNVs) in somatic cells and it is an active area of scientific research (e.g., Abyzov et al., 2017, Genome Res. 27: 512-523):
We estimate that on average a fibroblast cell in children has 1035 mostly benign mosaic SNVs.
Similar analyses in adults revealed no significant increase in the number of SNVs per cell, suggesting that a major fraction of mosaic SNVs in fibroblasts arises during development.
There are several answers to this question, depending on exactly what you mean.
Cells in the body will accumulate random mutations as they divide and multiply
The cells in the body that are not part of the reproductive line are referred to as "somatic cells". These cells are typically derived from a single fertilised egg. Every time the cell divides, DNA must be duplicated, and each time it is duplicated there is a chance that there will be a random mutation. These random mutations will be passed on to any cells which are later derived from this cell. Most work in this area is on single-nucleotide variations (see the answer from Martin Klvana) but there can also be insertions, deletions, and duplications as with other mutations.
Damage from radiation and other mutagens can cause DNA mutations
Cells are subject to damage all throughout their lives, and some forms of damage cause breaks or alterations to the DNA of the cell. There are complex repair mechanisms in place to deal with this damage but they are imperfect so over time this damage can result in differences in the DNA sequence of the cells in your body. This will increase with age.
Some cells have mechanisms for deliberate, random mutation
Certain cells in the body have evolved to be randomly different from each other. The most obvious example of this is in the immune system where random re-arrangments of specific sections of DNA are used to create a vast array of different antibodies.
Mobile genetic elements can activate and re-arrange through the body
The human genome, like that of other organisms, is full of transposons and other genetic elements that contain genetic machinery for relocating themselves within the genome. The activity of these elements in the somatic cells of the body can lead to differences in DNA sequence.
Not everyone has all their cells from a single fertilised egg
It can happen that rather than deriving from a single fertilised egg, a person is derived from two or more fertilised eggs, so that their body is made up of cells of distinct genetic origin that are effectively siblings to each other. This can result in large genetic differences between cells.
Furthermore, in women who have had children, it is common (found in around 50-75% of women) for some cells from the fetus to enter the mothers' body and permanently take up residence in a phenomenon called fetomaternal microchimerism. Again this can lead to large genetic differences between cells.
Differences in DNA go far beyond mere differences in sequence
All the mechanisms I have discussed so far will lead to differences in DNA sequence, however the DNA in a cell is more than a mere repository of sequence information, and its function is mediated both through chemical modifications, most notably DNA methylation, and interaction with proteins, most notably through the histones which mediate DNA folding in the cell, or through the selective inactivation of one X chromosome in the Barr's body if you are female. These modifications have profound effects on the activity of the DNA within the cell and shouldn't be ignored.
All of this means there is no single answer to your question since it changes with time, varies between individuals, and depends on exactly what you mean.