My character fell into boiling mud and I used misty step to get out. When I teleport am I still covered in boiling mud, and will I continue to take damage?
According to the description from Tales of the Yawning Portal p. 100 (or here on DNDBeyond):
Any creature that falls into the boiling mud takes 44 (8d10) fire damage at the start of each of its turns for as long as it remains in the mud.
The adventure book does not describe any mechanic for sustained damage once the creature exits the boiling mud. Of course, your DM is free to houserule otherwise, but if he doesn't it can be assumed that the mud does not continue sticking to you in an amount enough to cause further damage once you're out (no matter how you get out).
Now to address your direct question: there is no mechanic for what happens to things stuck to you when you cast misty step. Personally, I would rule that there would be a thin layer/traces of mud stuck to you afterwards (the same way that teleporting out of water might leave you wet), but not enough to do any significant damage.
The spell only teleports "you." In common D&D interpretation that means you and your currently worn/equipped items.
According to the DMG p246
For the purpose of these rules, an object is a discrete, inanimate item like a window, door, sword, book, table, chair, or stone, not a building or a vehicle that is composed of many other objects.
This would not include liquids of any form, and so they cannot be teleported (without being in a container).