I called the Australian Passport Office, as I couldn't find this information anywhere on their website.

They confirmed that passport numbers do indeed change both at a renewal and also at a passport reissue due to name change.

Obviously this only applies for Australian passports; I'm unaware of whether this is a general rule for other countries.

Yes the passport number changes upon renewal. The new passport will have it's own unique identifier i.e. its own passport number. This is a common practice. The old passport number will be invalidated and will appear as such when checked by border officers if you try to use it to travel.

As an authoritative reference, the British embassy in Bern confirms this:

Informative Note

Passport numbers change upon renewal

To whom it may concern:

This is to confirm that when a British passport is issued it carries a new number which is not related to the number of any previous passport. The British Embassy is not able to provide a letter or certificate confirming the passport numbers.

Yes, your passport number will change.

I have looked at my old and new passports (2 UK ones and 3 Australian ones) and the numbers are different for every renewal/replacement.

Additionally for me, I have had different issuing offices for each one, so that may be the reason mine have changed.

YES

I'm an Australian citizen and have had five passports, perhaps six, beginning in 1989.

Each time I've renewed my passport over the years the new passport has had a number different to the old one.

Yes, it will change.

A quick google search reveals it certainly does so for all major English-speaking nations.
I believe this to be true as a general matter of fact, however I can not find an authoritative reference at the moment.

However, Wikipedia says:

A standard passport booklet format includes the cover, [...]. Passports have numerical or alphanumerical designators ("serial number") assigned by the issuing authority.

So the passport number being a serial number, it's there to uniquely identify the passport, not the bearer, and thus should change with every re-issue of the passport.

As others have said, your new passport will almost certainly have a new number, and it is this number which you should record on your booking, so that it matches when you get to the airport.

However, note that you generally don't have to provide a passport number at time of booking, only before check in, because the airlines / travel agents know that you might not have it with you, and don't want to lose the sale.

If you provide the old details, they may query or charge you when you go to change them, but you should be able to tick that you'll provide them later. Then you can wait until your new passport arrives and fill out the correct details on a "manage my booking" page.

I am American. Things may be different in Australia but probably not.

When I renewed my passport, the new passport had a different number. However, I did not lose my old passport. Any visa in it are still valid. I can still use my old passport for many purposes. Book your flight with your old passport. Bring both passports to the airport. There will not be a problem with numbers because when they ask to see your passport you will give them the old passport.

Yes, the passport number will change.

The reason that airlines ask for passport information when booking flights is so that the airline can satisfy any pre-clearance requirements by immigration authorities at the destination. Typically, the information needs to be transmitted to the destination country authorities a certain number of hours or days before the aircraft departure. If you have a new passport, you should contact the airline and simply register the new information. Problem solved.