If you're favouriting a tweet to send a message to someone, eg "I liked your tweet" or "I've read your tweet" - carry on, but be aware that if this new thing rolls out to all users then this may be made more obvious to your followers (though it has always been public information, just not 'surfaced' in this way).
If you use favourites more as a 'saving for later' bookmarking kind of thing then do any of these instead. Instructions are for people using Twitter via a web browse.
1. Bookmark the tweet
Each tweet has its own web address. You can find it in its timestamp and right-click, bookmark the link, copy or save it into a new tab and bookmark it that way (this will depend on your browser settings).
2. Email it to yourself
Look for the "More..." option when is more visible when you hover over the tweet - emailing the tweet is an option there
3. Take a screenshot and save the image as a file
4. Storify it
Storifying is an imperfect tool for covert tweet-saving because if you publish the Storify and the person whose tweet you've saved has also registered with the site then I think they will be auto-informed by email and there's not much you can do about it, even if you avoid sending them the 'I've included one of your tweets in my Storify' tweets. However if you keep it as a draft it's nicely under the radar.
Twitter starts to change the central logic of its service (17 August 2014) The Atlantic
How to understand Twitter's new bad direction (23 August 2014) Computerworld (page three)
"Additionally, when we identify a Tweet, an account to follow, or other content that's popular or relevant, we may add it to your timeline. This means you will sometimes see Tweets from accounts you don't follow. We select each Tweet using a variety of signals, including how popular it is and how people in your network are interacting with it. Our goal is to make your home timeline even more relevant and interesting."
From What's a Twitter timeline? (date unknown) Twitter Help Center