Like it? Save it!
Do you like rereading your favorite fics? Rewatching AMVs on loop? Are there pieces of fanart you never cease to admire?
If a specific fanwork is a source of joy in your life—a comfort read on rainy days, an art piece that inspires you to improve your craft, a cursed YTP that you inflict on every new friend who watches your favorite show—remember that it might be gone tomorrow.
Links can break when people change their screen names or platforms change their inner workings. Fanworks disappear when people delete their accounts, take down their posts, have their posts deleted without warning, or get suspended. Platforms disappear when they can't sustain a profit or when people fail to maintain them. Sometimes you'll get a heads up before your favorite fanwork vanishes; often you won't.
So you might want to smash that motherfucking download button.
Any image in your browser can be downloaded to your computer. That includes fanart! Just right-click it, then click "Save Image As…", then press enter or click "Save". On Tumblr, you can download videos the same way ("Save Video As…", then enter or "Save").
Ctrl + S (or right-clicking anywhere, then clicking "Save Page As…") lets you download an entire webpage. This copies all the current text, images, formatting, etc. into an HTML file, which you can then open in your browser even if you don't have an internet connection, and even if the page is later changed or deleted.
Note that this only works with relatively simple pages, like blog posts. You can't use it to download YouTube videos—for that, you'll need youtube-dl or one of the many "youtube to mp4" sites out there.
If you manage to navigate to a URL ending in .mp3, you can use Ctrl + S to download the audio file.
AO3 has a literal download button which lets you save fics as PDFs, HTML files, or the e-reader file format of your choice. There's no limit to the number of fics you can download.
Obviously, if a creator has deliberately taken down their work, you shouldn't re-upload it to a public space. But there's nothing wrong with keeping a private copy for yourself. And downloading fanworks is actively good as a protective measure against corporate censorship and digital decay.