Does tourist visa (90 days) apply after a long-term visa ends in Schengen countries?

Exiting Schengen and reentering does not seem to be a requirement to make your stay valid.

[EDIT] However, I just found that the French Consulate in Sydney specifically says you should leave Schengen and reenter, suggesting to go to the UK :

If you want to stay in the Schengen space (for up to 90 days) at the expiry of your working holiday visa, you will have to leave France and the Schengen space and re-enter the Schengen area the following day as a tourist for 90 days within a 6 months period. You may leave the Schengen area (passport stamped at the border) by going to the UK for example.

[end EDIT]

As other answers show it, it is possible to spend up to 90 days inside the Schengen area for tourism purpose even if you had a long-stay visa before.

This is only experience and is not an authoritative source, but a friend (who does not need a visa for short stay) recently ended her stay on a student visa in France. She decided to stay for tourism afterwards. She did not leave the Schengen until more than 10 days after her student visa expired (with an exit stamp showing the date). She entered Schengen area and exited twice soon after, and never had any issue with immigration (who barely checked her visa the second time). She crossed the border at different places in Europe, making it relatively obvious she was traveling (and not working for example).

Therefore, even though the Consulate of France in Sydney says otherwise, as quoted above, it appears likely that exiting and reentering to have exit and entry stamps showing the stays are separate is not a requirement. It seems possible to have continuously a long-stay visa and a short stay (if no short-stay visa is necessary), but I would recommend to leave Schengen if possible. I suppose that any proof that you are not working (e.g. a document showing the end of a contract) would help, in case the border police is suspicious.

As an Australian citizen, you do not need (and could not get) a Schengen visa but you can indeed visit the Schengen area for 90 days, visa-free, after the end of your working holiday visa. The relevant regulation is the Schengen Borders Code and in particular article 6:

1. For intended stays on the territory of the Member States of a duration of no more than 90 days in any 180-day period, which entails considering the 180-day period preceding each day of stay, the entry conditions for third-country nationals shall be the following:

[…]

2. For the purposes of implementing paragraph 1, the date of entry shall be considered as the first day of stay on the territory of the Member States and the date of exit shall be considered as the last day of stay on the territory of the Member States. Periods of stay authorised under a residence permit or a long-stay visa shall not be taken into account in the calculation of the duration of stay on the territory of the Member States.

I had the same situation as you. I decided to leave the Schengen area when my work visa expired (hopped over to London for the weekend) and came back through Paris. I had no trouble at border control with getting an entry stamp, and the border agent hardly looked at my expired visa.

So I don’t know whether you can automatically get the 90 days, but at least in my case, I got the 90 days after a trip to the UK.

My relevant question on SE is here.

Update:
I have now left and entered the Schengen area twice just fine after my work visa expired. I have entered through both Charles de Gaulle and Orly airports (both in Paris), and the border guards flipped through my passport, saw the expired visa, and still stamped my entry.

I asked an Australian and this was their response:

From what I understand of the WHV program, a Working Holiday visa
‘removes’ that country from the rest of the tourist visa 90-day
restriction.

So if you will have spent the last 90 days of your WHV visa in the
Netherlands, then there shouldn’t be an issue with travelling through
the rest of the Schengen area for the next 90 days. I’m unsure as to
whether that strikes the Netherlands as unvisitable, however.

Also, transitioning off your WHV to the regular tourist visa may
require a run through somewhere like London to ‘leave’ the Schengen
area on your WHV so you can re-enter as a tourist – but again, I
cannot directly claim knowledge on this.

I would get in touch with the Australian embassy – they should at
least be able to help you with more certainty than I.

According to this (unofficial) website you can only apply for a Schengen visa while holding a residence permit up to 3 months before the permit ends, preventing exactly the thing you want to do. But that’s for the UK, which is not a full Schengen member. And you’re from Australia, so you don’t actually need a visa to stay in the Schengen area as a tourist.

Interestingly, Australians also don’t need a visa to stay in the (non-Schengen) UK, so a loophole might be to first go to the UK and then enter the Schengen area as a tourist.

But all in all this is a pretty special case and I’d suggest you ask the Australian embassy about this.

To corroborate what Paulinchen2 said:

From the Dutch Immigration and Naturalisation Service (IND) website:

If you wish to stay in the Netherlands for longer than three months, you will need a residence permit. Check the residence wizard for further information.

From the FAQ regarding residence permits (emphasis mine):

Can I also apply for a residence permit if I am staying in the Netherlands as a tourist?

This will depend on your nationality. Most people must first have a provisional residence permit [Machtiging tot voorlopig verblijf, MVV] before they can apply for a residence permit.
You can only apply for the provisional residence permit in your country of origin or permanent residence. The conditions for obtaining a provisional residence permit depend on your purpose of residence. In the Residence wizard select your purpose of residence, nationality and ‘longer than 3 months’. You can then find out whether you require a provisional residence permit and if so, how to apply.

And from the FAQ regarding holidays longer than three months:

It is not possible to apply for a short stay visa for a period exceeding 3 months. If you wish to stay in the Netherlands for longer than 3 months, you must apply for a residence permit. In order to be eligible for a residence permit you need a purpose of residence other than holiday and you must meet the conditions of the purpose of residence you have stipulated.

The wording and language seems pretty clear to me, although your exact situation hasn’t been discussed. So, unfortunately for you, the answer seems to be that it’s not possible unless you already have a Dutch resident permit (unlikely, as you only need it if you are staying for more than 90 days, and your current visa is valid for only three months).

I am 99.9% sure that it gives you no extra time.

The 90 days you refer to is the maximum duration of stay without a visa. As an Australian you can apply for a tourist visa in the Schengen Zone.