Only if the Twitter image is an actual PNG, which this wasn't; [it is a JPEG](https://pbs.twimg.com/media/FUxiOBaXwAMlK1w.jpg?name=orig). Because this was not a PNG, Furbooru wasn't parsing the original version of this post correctly and thus couldn't show it or anything.
The thing about Twitter is since it's a massive social network, it has to parse gigabytes of data from millions of users, so the site recompresses most images to make sure that the site can continue to work as quickly as possible and reduce crashes. This usually involves converting them into JPEG format. [This page](https://twittercommunity.com/t/upcoming-changes-to-png-image-support/118695) explains which images get converted from PNG to JPEG format on Twitter.