Let's assume the following:
An adult white dragon has been caught by a powerful group of LE wizards. Somehow they managed to trap it inside the depths of their underground laboratories, snaring it into a prison cell, perpetually harvesting scales, blood, teeth and whatever else they deemed useful.
Going by the description of suitable locations for a White dragon’s lair (MM, page 102) deep subterranean chambers (i.e. its prison cell) would fit its taste; at least if it had the choice to settle there.
Assuming the dragon can't escape, can't just choose to die or kill itself and "lives" in its prison for some extended period of time (years, decades, centuries?), would the regional effects characteristic to its lair manifest?
The region containing a legendary white dragon's lair is warped by the dragon's magic, which creates one or more of the following effects:
To me the wording of the MM leans towards the effects being a passive thing, but it's not clearly stated anywhere, far as I am aware.
Does a legendary dragon have to will the regional effects around its lair into existence or is it a passive effect caused simply by their presence?
The mere presence of a legendary creature can have strange and wondrous effects on its environment, as noted in this section. Regional effects end abruptly or dissipate over time when the legendary creature dies.
Since the text states "The mere presence" of the creature can warp the environment, this would suggest that these effects are passive. The environment is warped by the dragon's innate magical nature without them having to do anything.
The only description of a lair in 5e states:
A legendary creature might have a section describing its lair and the special effects it can create while there, either by act of will or simply by being present. Such a section applies only to a legendary creature that spends a great deal of time in its lair.
So, the exact nature of a lair is not an explicit game term. That means we take the standard English definition. A quick google search of "lair" results in the following:
a wild animal's resting place, especially one that is well hidden
a secret or private place in which a person seeks concealment or seclusion.
The prison sounds well hidden, and the dragon is resting there, so it does (at least technically) meet the first definition. Ultimately, the DM could rule otherwise, but if the dragon chose to treat their prison as their home, I myself see no reason why it wouldn't eventually be treated as a lair. And as a result, the regional effects would eventually apply to the area around the prison whether the dragon willed them to or no.