In principle, we can assume that any Maiar that can be "killed" is stuck in their particular form; otherwise, what would be the point in destroying their bodies?
This makes things somewhat easier, because those Maiar who have been killed is a relatively short list.
The big ones, of course. A possible reason for their being stuck in Balrog-shape is put forth in the Ósanwe-kenta, a manuscript related to the "Quendi and Eldar" essay (published in The War of the Jewels), which was first published in the 1998 issue of Vinyar Tengwar:
"Here Pengolodh adds a long note on the use of hroar by the Valar. In brief he says that though in origin a 'self-arraying', it may tend to approach the state of 'incarnation', especially with the lesser members of that order (the Maiar). 'It is said that the longer and the more the same hroa is used, the greater is the bond of habit, and the less do the 'self-arrayed' desire to leave it.'
[Melkor's greatest servants] became wedded to the forms of their evil deeds, and if these bodies were taken from them or destroyed, they were nullified, until they had rebuilt a semblance of their former habitations, with which they could continue the evil courses in which they had become fixed.
Basically, the spiritual beings become "addicted" to the forms they use, and over time it becomes harder and harder to give it up. The Maiar are more susceptible to this phenomenon than the Valar are, which somewhat explains why this happened so much more quickly to the balrogs than it did to Morgoth - he had to disseminate the majority of his power into Arda, while they just needed to forget to change out of their balrog suits after a long aeon of balrogging it up.
I will admit that this isn't a completely satisfying explanation; we wouldn't expect the balrogs to have used their physical form all the time, after all, which makes it unclear how exactly this process worked. It is, however, all we really have to go on.
This one we don't know. Shamshiel points out elsewhere that Huan is most likely a Maiar, but we know that:
- He is killed
- He doesn't change shape in situations where it would be useful for him to do so
Which leaves us a little stuck. One possibility is that he was something like the Wizards, bound to a physical form in order to fulfil a specific purpose in Middle-earth. However, this would seem to be contradicted by the note Shamshiel cites, which says that Huan "robed himself":
Living things in Aman. As the Valar would robe themselves like the Children, many of the Maiar robed themselves like other lesser living things, as trees, flowers, beasts. (Huan.)
History of Middle-earth X Morgoth's Ring Part V: "Myths Transformed"
Which implies one of two possibilities:
- Like the balrogs, he was simply stuck in a form he'd been using for thousands of years
- He kept his wolfhound shape voluntarily throughout his time in Middle-earth
There's no evidence for or against either, but personally I think the first is vastly more likely.
I put a question mark because we don't know if Melian was bound to her physical form. If we assume that Huan was, it seems likely; she was in that form for longer than he was in his, after all (so far as we know, anyway).
There is some support for this from the text, but it's a little shaky:
Melian was of the divine race of the Valar, and she was a Maia of great power and wisdom; but for love of Elwë Singollo she took upon herself the form of the Elder Children of Ilúvatar, and in that union she became bound by the chain and trammels of the flesh of Arda.
The Silmarillion III Quenta Silmarillion Chapter 22: "Of the Ruin of Doriath"
One possible reading of this passage is that Melian has literally become bound up in the physical matter of Arda - which seems to imply that she's become stuck in her body.
Further supporting this interpretation is Ósanwe-kenta, a manuscript related to the "Quendi and Eldar" essay published in The War of the Jewels. In this manuscript, Tolkien writes (bold is my emphasis, italic is his):
Pengolodh also cites the opinion that if a "spirit" (that is, one of those not embodied by creation) uses a hröa for the furtherance of its personal purposes, or (still more) for the enjoyment of bodily faculties, it finds it increasingly difficult to operate without the hröa. The things that are most binding are those that in the Incarnate have to do with the life of the hröa itself, its sustenance and its propagation. Thus eating and drinking are binding, but not the delight in beauty of sound or form. Most binding is begetting or conceiving.
"We do not know the axani (laws, rules, as primarily proceeding from Eru) that were laid down upon the Valar with particular reference to their state, but it seems clear that there was no axan against these things. Nonetheless it appears to be an axan, or maybe necessary consequence, that if they are done, then the spirit must dwell in the body that it used, and be under the same necessities as the Incarnate. The only case that is known in the histories of the Eldar is that of Melian who became the spouse of King Elu-thingol.
However, we learn later that she returns to Aman; how she could have done this while being stuck in one form is unknown.
As I said before, any Maiar-like creature that can be killed can, presumably, be said to be stuck in a physical form; so we can extend that list to include an unknown subset of the following list:
- Orc-form Maiar
- Possibly the Ents
- Possible the Eagles
- Possibly the Werewolves
- Possibly the Giant Spiders
- Possibly Dragons
If we accept that "staying in one form for a long time" makes you become stuck to that form, then we may have to include Tilion, the Moon; although we don't know if he had to incarnate in order to fill his role1. If he did, then presumably he's been stuck in that form since the start of the First Age.
We may also have to include other named Maiar (such as Eönwë, or Ossë); the fact that they were known to the Eldar indicates that they had some consistent form that they used, and Tolkien's quote above suggests that frequent use of the same form also causes one to become "stuck." However, we cannot confirm that this happened to any of the other Maiar; we're never told so, and we just don't have enough information on how the process works.
1 Though we know Arien, the Sun, did shed her physical form in order to become the Sun. Thanks to Mithoron for correcting me on this