During mitosis the genetic material in the cell is replicated to produce a copy of the genome for each resulting daughter cell. Due to the nature of the process, the ends of the chromosomes are not completely replicated, resulting in a slightly shorter copy of each chromosome after each round of replication.
Telomeres are extensions to the end of chromosomes that prevent damage or loss of genetic information during cell division. Telomeres are not replaced (in 'normal'/somatic cells), which gives rise to a replicative lifespan; the number of times a cell can divide before permenantly leaving the cell cycle (known as cellular senescence).
This is generally viewed as an anti-cancer mechanism to protect against errors creeping in to the genome through many cell divisions. In order to become cancerous, a cell must first overcome its replicative lifespan [ref.]. This is achieved by activating the (normally inactive) telomerase enzyme that extends the telomeres - embryonic stem cells are one of the few cell types that normally express this enzyme, so they have unlimited replicative potential - a very important trait for stem cells.
So a rapidly proliferating cell would indeed 'use up' it's telomeres before a different cell type. The cell would then either enter a state of senscence (permenant cell-cycle arrest), or apoptose. There are a lot of factors governing which outcome is realized, but the senscent cell population increases with age, and is proposed to contribute to many aging phenotypes (there is a recent fascinating study published in Nature where the authors remove all the senscent cells from aging mice, and the mice actually get healthier! Can't wait for studies that relate to human aging and senescent cell clearance (ref.)).
I have elaborated on the function of telomeres in the context of organismal aging in my answer to this question.
