I am having this exact same problem as I type this, I can refine dilithium on my Federation Engineer character, but I cannot refine dilithium on my Federation Tactical Character or on my Klingon Defense Force character.
While I have no problems manually refining on my Fed main (nonzero balance) I have noticed the automatic refining fails a lot, or it refines zero, even if I hadn't refined manually that day, with lots of ore sitting around. I'm not the only one in my fleet with this problem too.
My KDF is an Orion "entertainer" that never leaves Drozana... so... no problems there.
The common theme seems to be that people who have this bug tend to have a zero balance in Refined Dilithium.
From what I can tell this problem is caused by reaching your daily refining capacity. The 'you can refine [x amount] dilithium today' counter doesn't reset automatically. Instead, it resets when you refine dilithium for the first time each day. So, if you refine 8000 in one day, you get stuck with disabled 'refine' button and can't reset the counter ever again.
I got response from support ... they are a bunch of ******* ...
Then i took the matters in my own hands.
I got two stations from which i log on. I observed the problem from one of them so desided to check if i log on on the second computer it'll be the same.
I t seems that i can refine dilithium from the second station/computer/client but not from the first one.
So i think this problem has someting to do with the client (locally). That said - a simple client reinstall should remedy the problem.