According to W3Cschools just 17% of users still use IE6.
Source
Also their statistics show that just 4% of users have a resolution of 800x600 or lower.
Source
At what point do I neglect users that refuse to update in order to improve the experience for rest of the users?
How much do you think the content of the web page effects this threshold?
At the point where you are reaching a sufficient portion of your target audience that the cost becomes too great to make up for the additional little bit you gain by supporting a lessor-used browser.
For most sites, 17% is too much to drop for the little bit of work required to support it.
But IE5 is small enough, and the compatibility issues large enough that it doesn't make sense to support it.
If you're running a site oriented to a technical audience you might be able to drop ie6 as very few of your target audience would be turned away, and the additional work to support them is greater than the value they represent to your site.
Regarding screen resolution - keep in mind that more and more users are experiencing the internet through mobile devices. This is a question that you can only answer by looking at your content/application, and your userbase, and making a judgment call.
For the most part, big sites support a wide range of resolutions and devices transparently, but you may only need to support one once you look at your audience and application.
If you use a cross browser toolkit like GWT, dojo and so forth, you should not have to neglect anyone. That's exactly what they are made for.
I support IE6 and 1024x768 because that is what most of my target audience (education) is using.
I suggest gathering some statistics before you decide what you will/won't support.
You should cater to your audience as much as possible.
If your site generates revenue from clients, then
Sum(Revenue from 86% users) > 3 * Sum(Revenue from 14% users (who use IE6))
else
every client is king.
I guess it does depend on your site's audience and not on stats of w3cschools. According to the site Firefox is 46% (march 2009). For my site, 63% of the trafic is generated by ff. So you can assume quite a difference for your own site.
If you make money with the site, look at the numbers and depending on your user conversion rate decide if it is worth making the site available for ie6 and the expected return on investment.
Keep in mind that many users don't browse with the browser maximized so screen resolution is a relatively useless statistic.
If you don't make money from your site, I would definitely not do any extra work to support IE6. If it were a technical site, I wouldn't even bother with IE at all, just stick to standards (of course, it probably would work in IE 7/8).
If you make money from your site, consider this. Would you be okay with 17% less profit?
I wouldn't neglect IE6 users yet. Not all users will necessarily be in a position to upgrade, e.g. some users may be accessing your site from their workplace where the IT department mandates a particular version of IE.