If you are in a room with cameras in it, is it like the people looking through those cameras are in the room?
There are many cases in Halacha which involve having someone in the room. For many of them, an actual person in the room is required; a video camera is not enough. But if we just need to be sure about something, a video often works.
For many laws of kosher, the requirement is to be sure that the food is kosher; a video camera may be good enough for this.
- Chalav Yisrael requires that a Jew watch the milking (though he can step in and out every few minutes); R' Moshe Feinstein feels that "knowing for sure" is as good as watching, so I think he'd allow a video camera. I don't know if other rabbis require actual watching, or if video would count too.
- Kosher meat requires constant supervision; I've heard of some rabbis today talk about if having a video camera might work for this.
I doubt a video camera is enough to prevent the problem of yichud.
If we can see someone stealing money on the video camera, that's enough proof to make them pay it back. (There's a story where R' Saadiah Gaon used science about 1000 years ago to solve a question of who owes money.) I'm not sure it would be enough proof to give the death penalty for a murderer, though.
Sometimes we need witnesses not just to prove something, but to make it happen in the first place. A Jewish wedding doesn't work unless two witnesses watch him give her the ring -- otherwise, they're not married! There's some discussion about whether the witnesses need to "see it" or just "know it"; but we would probably require that the witnesses actually be there, not just on video.
As far as davening with a minyan goes, I know they talk about it. The Gemara says that the big shul in Alexandria (Egypt, about 1800 years ago) was so big, they had to raise a flag so you knew to say "Amen" because people couldn't hear.
There are lots of mitzvas that you can either do yourself, or listen to someone else do (like davening or brachos or reading a megillah). The rabbis talk about whether listening on microphone or radio is good enough (Rabbi Ovadiah Yosef's grandfather said no about the megillah). But I can't think of many mitzvas that you have to SEE something. It would probably be a similar discussion about whether seeing it on camera works.
Those are a few cases I can think of; did you have something else in mind?