Sounds like a flute combined with a lot of EQ, leaving only the mids with no attack. This is accompanied with the occasional string pluck of what seems like a guitar. Same EQ, which makes the two instruments blend together in an etherial kind of way. Slap some spring reverb on top.

EDIT: The strings sounds like the overtones of a half acoustic guitar. I think the vibrato is artificially applied. It's kind of possible to make that effect by slapping the whammy bar, but not in a way that'd make it fade in like that. Maybe if you lightly touch the already vibrating string in the right spot you could achieve something like that, but I'm unsure if it'd work, or if it'd create an overtone on an overtone.

UPDATE: Yup, I just tested it. Those are definitely overtones of a guitar. I couldn't replicate the vibrato, so I'm pretty such that's been applied in post.

As a tangent, I watched all of House from start to finish when each episode was released by EZTV. As a result, I only ever heard the american intro despite being European. I didn't even know other intro songs existed until now. I guess they didn't have international distribution rights to Teardrop.

guitar string harmonic run through a ring modulator, delay and EQ.

Ring modulator https://youtube.com/shorts/t9Z2GELc-hw?si=uvv-UQ-My4QzCbNm

https://www.looper.com/322092/the-truth-about-the-house-m-d-theme-song/

It sounds like a dial tone to me, with a lot of fuzz. Could be a pager, as doctors used to use them a lot.

Sounds like a buzz and a sine wave mixed and lots of reverb. It reminds me of a flute. The truth is that a lot of these sounds are usually synthesized, it is the signature style of Massive Attack. As a result it shares a lot of similarities with the kind of simple analog sound makers used in hospital equipment. It doesn't sound sampled to me, at least, but I could be wrong, it has a lots of things layered on top to say for sure. If anything, it reminds me of the stereotypical code blue alert used in drama shows. Though I have never encountered such a thing in real life, usually those things are communicated verbally without any sort of alarm. Hospitals actually avoid alarming or excitedly modes of operation, preferring calm and collected action.

Sounds like a string or piano, with a lot of effects/manipulations.