Failing to get to some, but not all websites is more likely to be an ISP issue. In this instance two websites were mentioned, support.mozilla.org and www.liveleak.com. The originator of this question must have been able to get to support.mozilla.org in order to post the comment that they were having trouble getting to support.mozilla.org. Just now for me, both websites come up in the usual amount of time.

An example of an ISP issues, anyone using AT&T DSL in my area was getting timed out like the original owner of this question with support.mozilla.org except, in our case, it was anything going to yahoo.com. The connection somewhere in the ISP backbone was misrouting or maybe not routing traffic to yahoo.com. To take the browser (firefox) out of the equation, try a command outside the browser. From a command line enter the command tracert www.liveleak.com [on an XP system] or traceroute www.liveleak.com [from a linux system]. If it goes not show all the node-steps between you and the final destination, then somewhere traffic from you to them is getting hung/lost

Both error-messages results when trying to websites other than liveleak and mozilla.

For websites that give the error-messages originally listed in this question: - Are they accessible on the same system outside firefox, either through a tracert or another browser (ie or chrome) within a few minutes of their failure on firefox?

if not, - are they accessible on another computer

Please, rule out the possibility of a problem with routing from your location, which would need to be addressed with your ISP

The websites that Firefox gives the error-messages are always successfully accessed without issue via Chrome and Internet Explorer

Happening in Firefox, not in Chrome or IE on same system so, more likely the problem has to do with firefox or your specific install (profiles, configuration etc). No mention was made as to your version of firefox other than it's Windows 10.

If the problem is actually within Firefox, a fix would need to be created. That fix would be included in a future update to firefox. In short if you're not at the current version of firefox, then upgrade and try again. Beyond that, a bug report would need to be generated with enough information and assistance so that others can reproduce the problem then it's a matter of developing a fix. Not all problems, are quickly fixed. An issue that hits my installation has been opened somewhere between 7 and 10 years. Another issue I reported was fixed on the next incremental update as it caused obvious issues for many people.

It's also possible the problem is something in the way firefox is installed on your system. Maybe someone with experience in your environment (Windows 10?) would be in a better position to assist.

Start Firefox in Safe Mode {web link} by holding down the <Shift> (Mac=Options) key, and then starting Firefox.

A small dialog should appear. Click Start In Safe Mode (not Refresh). Is the problem still there?

There is security software like Avast, Kaspersky, BitDefender and ESET that intercept secure connections and send their own certificate.

https://support.mozilla.org/en-US/kb/firefox-cant-load-websites-other-browsers-can

https://support.mozilla.org/en-US/kb/firefox-and-other-browsers-cant-load-websites

https://support.mozilla.org/en-US/kb/secure-connection-failed-error-message

https://support.mozilla.org/en-US/kb/connection-untrusted-error-message

http://kb.mozillazine.org/Error_loading_websites

https://www.bing.com/search?q=web+site+access+denied

Is the problem still there?

Yes:

Hmm. We’re having trouble finding that site.

We can’t connect to the server at www.facebook.com.

Did you check out the links I posted?