As with most things, it's complex and depends on context and you just don't say it the way you'd expect.
If you're asking for pizza at a pizza restaurant (we'll get to tea later), you say:
I'd like some pizza to go please.
'To go' is a modifier not exactly of the pizza itself but the order. 'To go' means to package it all up so it is easy to carry to my car, like in a bag, as opposed to a tray which is 'for here'
If you're calling on the phone, you say:
I'd like to place an order.
Is that pick up or delivery?
(which means are you going to go there and pick it up yourself, or do you want it delivered. Either way, it is the entire order that is considered 'takeout' (which is the preferred word over 'takeaway' in the US.
So when you get it home, and someone asks about the pizza in your hand, it might go:
Is that takeout or did you make it yourself?
You could say 'takeout pizza' but more often than not, you'd just say 'takeout', similar to you might respond to "Is it a big pizza?" with just "It's big.". Though you definitely would not say order it using 'I want some takeout pizza' (pragmatically it seems weird), it is normal enough to hear something like "That pizza shop has takeout pizza.".
But now to 'takeout tea'.
And then there's cultural reality. In the US, if you're getting food from a place that allows takeout vs eating there, it is just not a place to get tea. You'd almost always want to be sitting anyway. I suppose you could order a tea instead of coffee at a McDonald's drive-thru, but then it is necessarily understood to be takeaway.
So to summarize, no, you do not say "Can I have some takeout tea??". Instead you'd say
"Can I have some tea to go?"