Yes, definitely ! The expression level of genes (as measured by their corresponding RNA and/or protein levels) are constantly changing. Many different things can affect expression levels, and significant changes can occur in periods of time only seconds long.

http://en.wikipedia.org/wiki/Regulation_of_gene_expression

In an individual cell, mRNA levels can vary widely due to bursty transcription. This leads to high variation in mRNA expression levels in a single cell over time and across populations of cells of the same type.

See:

https://en.wikipedia.org/wiki/Transcriptional_noise

https://en.wikipedia.org/wiki/Transcriptional_bursting

There's a big difference between a single cell and a population of a single cell type. Microarray and RNAseq technologies generally look at mRNA expression in large populations of cells. This has an averaging effect, and the mean mRNA expression for the many thousands of cells required to get enough mRNA for analysis will be relatively stable.

For given cell types there should be many genes whose expression levels remain at or very near zero, however...

As for levels of protein, I've not seen references to that but I would expect them to be somewhat more stable.

The expression levels of the genes will change because the cell responds to its environment, and the environment will be different at two different points in time. Every gene is like a dimension in your experiment, so you might think of the cell a very complex set of variables which can respond to any environmental signal (including the passing of time). Any given gene can change for some reason or another (which affects other genes, which then change, etc.).

That said, there are relationships between genes that will have some consistency. Thus if you measure the gene expression profile of a region of skin, it will be reproducibly distinct from a profile measured in brain, and will be unique enough to characterize a type of skin in general. However, even that profile will show variability from person to person, and for the same person at different points in time. So the expression levels of genes is highly variable, but there are ensemble characteristics that can be observed.

I'd integrate the three responses from Stu, seidel and gawp, minmus the wikipedia references, and add in cell cycle oscillations as yet another force which leads to variation in transcription over time in both single cells and a population of cells.