I was searching some stuff about Tiamat on Google and found your question. It's been a while since you asked and my answer will not really solve the matter, so I don't know if it'll be useful, but it's still a valid reference, so here it goes.
The 5e DMG, on the section about the Planes of the Multiverse, specifically on the paragraphs about Avernus (p. 64), states that:
"Tiamat, the Queen of Evil Dragons,
is a prisoner on this layer, ruling
her own domain but confined to the
Nine Hells by Asmodeus in accordance
with some ancient contract (the terms
of which are known only to Tiamat and
the Lords of the Nine)."
I've never seen any other mention about this contract (neither on previous editions nor in other books of the 5th). So it turns out that the sourcebooks kinda leave the matter open to DM's imagination.
As others have said already: Tiamat is clearly imprisioned (not in actual chains and stuff, but more like enslaved/serving) and can't get out of Baator without aid.
Baator is her home Plane now and she resides in Avernus serving as an advisor to Zariel (the first Lord who lost her porsition to Bel; then at some point Tiamat was made Lord of Avernus, but she really sucked at it so Bel had the position back; and for some unknown reason Bel was demoted and Zariel was made Lord of the First once again... uff, it's complicated, but it's something like that). Tiamat is also the guardian of the gates to the second layer of hell, Dis.
She's agreed to serve Asmodeus and not to leave the Nine Hells, so the contract must grant her some benefits (maybe she can draw some of Baator's magical power the same way Devils do). Or perhaps she needed some aid/protection ages ago and Asmodeus agreed to "rescue" her if she signed the contract.
The thing is that the terms (and the history) of this contract are not really set, so a DM can pretty much create anything to suit his campaign! :D
Hope it helped o/