Brewing With Tea

No point letting it sit so long, IMO. Think about brewing tea. If you used hot water, you would let it sit 5 minutes. If you used warm water from the tap, maybe a half hour. If you used cold water from the tap, maybe overnight in the fridge.

How much? I don't really know, but you probably want the vodka that you add to have no perceptible vodka presence, so make it concentrated. I might try making some regular tea at high strength, maybe 1 bag in a quarter cup of water. Start adding it dropwise to your beer. How many drops does it take to reach a tea flavor that you like? Does the beer taste watered down when you get there? If so, it needs to be more concentrated.

 

Tea does not like being boiled. It's a similar thing that happens when hops are boiled. Bitterness via the breaking down of its essential science language parts.
You really want to pay attention to exposure time as well if you're doing it as a steep/whirlpool addition on the hotside. Too much exposure extracts more than you want from the tea.
All that said.
I'd personally make a tea cold in filtered water, and add it as a secondary, or as part of the primer for bottling. Like hops for ipa freaks. Freshness matters.