Stem cells persist because they have the capacity to self-renew. At birth you will have a finite number of each stem cell type (e.g. hematopoietic stem cells that make blood, gut epithelium stem cells that make gut lining etc). The general picture is that as tissues require replenishment, a stem cell divides. One daughter cell starts to differentiate (and divide) into mature cells required by the tissue, whereas the other daughter cell retains stem cell function.
So a stem cell is typically defined as a cell with the capacity to: (i) self-renew and (ii) to give rise to one or more cell types in a tissue. A hematopoietic stem cell can give rise to T-cells, B-cells, neutrophils, red blood cells, platelets etc. The ability to self-renew means that, in general, stem cells don't 'run out' during a lifetime.
[ Its important to note that to define a cell as a stem cell, scientists use strange hybrid of: in vitro or animal assays + cell surface markers + location in the tissue + measures of 'having the capacity to make continued contribution to new cells'. So lets just say that stem cells are only ever as well defined as our choice of assays combined with the interpretation of those results. ]
I mention the challenge of defining stem cells because, in recent years as assays and technologies improved, cells such a oocytes (eggs) or adult neurons, which were once thought to be finite, have been shown to have be replaced/renewed, presumably by some 'stem cell like' cell(s).
As noted above, stem cells are usually involved in the replenishment of a particular tissue type. So liposuction, which will remove fat cells, should not affect the numbers of hematopoietic or neural stem cells and so on.
Any claim of clinical benefit of a 'stem cell treatment' using liposuction should be backed by clear data.
A longer answer would get into:
(i) the concept of 'plasticity' where the classical view - a fixed pathway between a stem cell giving rise to one daughter cell that gives rise to mature cells and the other returns to a stem cell state - is open for debate, and
(ii) the idea that since stem cell definitions are only as good as our assays, there may be potential plasticity even among stem cells that we cannot 'detect' yet.