I’m an Australian resident with a UK residence permit. Can I stay in the Schengen area for more than 90 days?

I assume that you're actually a citizen of Australia, not just a resident, since that seems to be the interpretation under which the rest of the question makes the most sense. If that assumption is incorrect, please leave a comment on this answer, and I will change it as appropriate.

The UK residence permit has no bearing on your ability to spend time in the Schengen area. (Indeed, holders of residence permits for Schengen countries are technically limited by the 90/180 rule when it comes to spending time in other Schengen countries.)

Does anyone actually check? Sometimes they do. They maybe more likely to check when you leave than when you enter, since that will be their first opportunity to catch you if you overstay.

I suppose they don't count all of everyone's Schengen stamps every time they cross the border because that would be too labor intensive, but neither do they have a policy of routinely ignoring the issue. The potential negative consequences are severe, so you should think very carefully about whether you're willing to risk it.

If you want to spend more than 90 days in the Schengen area, you should either plan to spread them out sufficiently or look very seriously at whether you can get a working holiday visa or other long-stay visa for your desired destination. This will give you 90 days out of 180 in the rest of the area.