The differences, I think, are mostly aspectual rather than semantic. That is,
- be up against tends to be used to describe the state of wrestling with a difficulty—“I’m up against something here I don’t know how to handle.”
- come up against tends to be used to describe encountering a difficulty. However, it is often, perhaps usually, employed in the present perfect—*We’ve come up against something we can’t handle, and we wondered if you have any ideas”—which carries the notion of a state created in the past but enduring into the present.
- run up against is not significantly different from come up against. I would tend to favour it in contexts where a difficulty was encountered unexpectedly, arresting progress abruptly, as if it had overtones of “running into a brick wall”; but that’s quite possibly only a personal preference.
None of these is written in stone; they are tendencies only, and I think you may safely use any of them in any context and they will be perceived to mean exactly the same thing: you’ve got a problem.