It's very hypocritical to slate IE for its lack of standards compatibility when we still nowadays only test for the popular browsers. It can sometimes cause accessibility problems if your HTML is very badly munged.Otherwise, there aren't really any practical reasons to worry about it.
As a diligent developer I am sure you will retest as browsers get updated but there is a window between the update and you testing (remediating).
:-) I find that validation is a matter of principle more than what is required. If you are having trouble with CSS, making sure your HTML and CSS validate is a really good diagnostic step.
Just because your site works with this crop of web browsers, there is not guarantees you will be good with the next if you are not standards compliant.
Lets be honest web browsers will often be updated many times within the lifespan of a web site.
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I always validate my web pages, and I recommend you do the same BUT many large company websites DO NOT and cannot validate because the importance of the website looking exactly the same on all systems requires rules to be broken.
In general, valid websites help your page look good even on odd configurations (like cell phones) so you should always at least try to make it validate.
Problems that visitors have when they view pages with different browsers may often be due to bad HTML and not an issue with the browser.
Furthermore, some browsers are less forgiving of bad HTML than other browsers like Internet Explorer (which can be very forgiving).
I have been developing websites for a couple years now and I almost never check if my pages are valid html and css.