If I disable third-party cookies in my web browser is this likely to cause me many issues when browsing the web? In particular, do they have common uses other than advertising?
It's possible that you might have issues if you disable third party cookies.
In other words, disabling third party cookies might render some web page elements useless, if they are loaded from a "third party", that is, from another site than the page a user is visiting.
There's some interesting discussion of this on the Mozilla bug-tracking software, in this request to disable 3rd-party cookies by default in future versions of Firefox. One quote from a Firefox dev:
However, Safari (at least on mobile) disables third-party cookies by default, and the internet doesn't seem to be broken from my iPhone. (Although there are ways for advertisers to get around the restriction, and even Google has been caught doing so.)