Should I use dried or fresh chilli/ginger in chai tea?
It depends on how they are dried.  Though if they're properly dehydrated or freeze-dried then you're just rehydrating them in the tea and they'll taste ok.

Ginger: I prefer fresh ginger and find that there are some taste notes that aren't present in the dried tea, it makes more pleasant with fresh ginger, but it's more work.

Chili: While I haven't tried fresh chili in tea, I'd imagine dried would work better and you wouldn't get interference from oils in the chili.  Also maybe easier to control the amount.