I have just read a paper from cancer research in which they do pathway enrichment using up-regulated or down-regulated differential expressed genes. My understanding about functional enrichment is that a biological process may be perturbed by both up- and down-regulated genes from the mRNA expression level. So is there any biological hypothesis to perform the functional enrichment using only up-regulated or down-regulated genes?
Yes, although life is complicated and I think it's rarely a good idea. Many times one might hypothesize that a given process will be affected by an intervention, and the prediction will be for an increase or decrease in the activity of that process. For example, radiation exposure would be expected to result in short-term increases in expression levels of genes involved in radiation response. However, most interventions result in complex and sometimes counter-intuitive responses. Radiation exposure in cell lines results in increases in DNA damage repair genes but decreases in mitotic gene expression as the cells stop replicating. Decreases in gene expression level in a negative regulator of a process can have the same effect as increases in a positive regulator. Looking at only up-regulated genes misses a lot of the picture.
Just to add to David's response, you might consider what counts as a biological hypothesis. It could be something as small as looking for a single gene, or large and complex like a developmental pathway. If you're looking for any potential target gene of transcription factor X, and you find some condition in which X is over-expressed, the up genes are the only ones with a direct hypothesis, and you might not have a way to understand what the down genes mean (yet).
On the other hand, looking for complex phenomena (pathways) is more difficult and complicated, but then the term enrichment means just that - more concentrated than expected by chance. If there are 20 genes involved in some process, and 5 of them show up in one end of your spectrum, that may be enough to bring that process to your attention. It doesn't matter if 5 others from the same process are also found expressed in the other direction, and the remaining 10 do not show up at either end. What's important is that the pathway moves into your field of view - it doesn't mean all the genes involved in a process are doing the same thing. So to answer your question, yes, gene expression is often used as a definition of biological processes, and thus the up or down gene sets identified in a given experiment can be tested for enrichment of genes that make up some process.
Good answers by the others; I agree with what is written. I'll add some other thoughts. One other way to look at this is the intense interaction/reaction between organism and environment. To survive, an organism and its cells need to be able to sense and respond to its environment - be that external, or micro in the vicinity of given cells.
This in some ways is a broader view of what the others have written - environmental stimuli affects gene and protein expression with many of the affected genes showing coordinated biological activity, ie, assignment to biochemical, developmental, etc pathways. Coordination is key, as this is fundamental to homeostasis, which keeps the organism and its cells in balance as the environment changes.