: How does the tonic solfa work in minor keys? I'm curious as to how it works it minor keys specifically melodic and harmonic minors. I'm assuming it is different than the regular Do-Rei-Mi.

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There are two ways to use the do-re-mi note names:

Absolute note names
Relative note names

If you use them as absolute note names, you just call C do, D re etcetera, and if you encounter alterations, use the naming suggested by Babu (although he suggested it in the context of relative note names).

If you want to use them as relative note names, Babu suggests to call the tonic Do. I don't like this, I prefer to call the tonic of major keys Do and the tonic of minor keys La. A few advantages this has:

There is no difference in note names between a key and its relative. The natural minor scale contains exactly the same notes as the major scale, it just starts from the sixth note of the major scale. (This means that if a piece switches between the two - which often happens - you don't need to think about naming.)
You need fewer accidentals then with Babu's method. You would need them only for the harmonic (...-mi-fa-si-la - I use "si" for raised sol, "ti" for the note below "do") and melodic (...-mi-fi-si-la-sol-fa-mi) minor scales.
It results in a distinction between minor and major keys in the very naming scheme, while all keys are the same. This reflects how it is in singing, where it doesn't matter from which key you start (barring range issues), but it does matter which mode you use.
This system is easily extendable to church modes, you can just say Dorian starts on re, Phrygian on mi, Lydian on fa and Mixolydian on sol. (Aeolian on la and Ionian on do as discussed, and if you want to be really pedantic, Locrian on ti.)

The relative note names work because there are fixed intervals between the notes: do-re, re-mi, fa-sol, sol-la is one tone, mi-fa and ti-do a semitone. (Of course alterations lower or raise the altered note by a semitone. I'm leaving the distances including altered notes as an exercise for the reader.)

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Solfege has names for altered notes. From fixed Do as C, we have:

C# as Di,
D# as Ri,
F# as Fi,
G# as Si, and
A# as Li,

We also have:

Db as Ra,
Eb as Me (pronounced "May", not as the English word "Me"),
Gb as Se,
Ab as Le, and
Bb as Te.

I'm told (through Wikipedia's Solfège Page) that there are names for double-sharp and double-flat notes, but I've never used them and cannot comment on their general acceptance.
In fixed Do, conceptually, it's a straightforward conversion; C# is Di, so you always sing Di when you see C#. It's a little trickier to sing than movable do (a.k.a. tonic sol-fa). In movable Do, you only alter the notes of the scale which are flatted compared to the parallel major. So a natural minor scale would be:

Do-Re-Me-Fa-Sol-Le-Te-Do

A harmonic minor scale would be:

Do-Re-Me-Fa-Sol-Le-Ti-Do

An ascending melodic minor scale would be:

Do-Re-Me-Fa-Sol-La-Ti-Do

And descending would be the same as natural minor.