Ttdfjt

There is a vocal exercise in which you sing a do-re-mi--fa-sol-fa-mi-re-do-do-sol-do. You do this in many keys as possible.
Is it good to do this with hunming? What is the purpose of doing this exercice with humming?

Not only are such exercises good for getting your voice in tune but for learning to get the correct resonance and support for all types of syllables. The specific do-re-mi- etc sequence you provided is just one of several (hundred or thousand) that one can come up with.

The same sequence can be practiced using the following:

Lip trill

Staccato (on "pa" for example)

Long syllables (Va, Fa, Ma, My, Moi, etc), with different vowel endings

On Ng (difficult to explain, but this is close to humming with the mouth open and tongue pulled back)

And humming

In general it is not desirable to close or cover the mouth when singing but one cannot avoid changes in mouth shape when singing. One point of these different exercises is to teach one to not lose the support or resonance when transitioning from one syllable to another. Also, the internal feeling of the resonance is different for each of these exercises and for vocalists it is important to be able to feel and identify the resonance in the sinus. Humming does support this effort.

"Warming ups" together with solfege (the relative doremi system) will help you to get used and find the root and tonic. There are many different vocals and consonants to train your voice, resonance rooms of head, nose, breast, to control your throat and tongue, your breathing etc.

It makes a lot of sence to practice these exercises together with the relative key of doremi. You can also start with so-fa-mi-re-do - somido (and always moving a half tone higher), singing uh, oh, ah, sümsüwü etc. or adding a consonant mia, mia, mia, mia, mia....
nia, nia, nia, etc. or ping,pong, etc.

Before operating muscles, it's always a good thing to warm them up by exercising them. There are lots of muscles involved in singing, mostly little ones in the throat, etc, and a big one operating the diaphragm.

Humming uses far less air and air pressure than singing with open mouth, so as a starter, it is a more gentle warm up than 'proper' singing, albeit with single syllable words - lah, pah, bah - which always leaves me feeling sheepish.

So, there's nothing wrong with a good hum to start, obviously the solfege words won't be forthcoming, but pitch is still there, and with many, many different pitch exercises to be chosen from, why not start there?