I love following writing prompt blogs and accounts on social media because they really get my creativity flowing, my favorite being Writing Prompts on Tumblr and Instagram. I've never actually incorporated any of the ones I find into serious pieces, but if I were to, would I be violating copyright? Are writing prompts put out online subject to copyright or intellectual property laws? And if I did, hypothetically, base a work that I wanted to publish around a writing prompt from, say, Tumblr or Instagram, would I need to get permission from the original poster?
Yes and no. The thing about copyright is that applies to the form of expression. This means the actual words used to express whatever the writer is trying to say. It would be an infringement of their copyright if you published or posted their actual writing prompts without their permission (generally).
I say generally because whoever wrote a writing prompt might, please this is not definite only a possibility, be agreeable if you republished or reposted their writing prompt elsewhere on the internet especially if you gave them full attribution as the creator.
Most of the writing prompts on, for example, Tumblr that you linked to, are actually more or less ideas or suggestions for potential stories. Ideas themselves are not copyright. This makes it possible and not illegal for you to take a given writing prompt and build a story based on it. Permission isn't required. Although, at some stage you might acknowledge the source of your inspiration as a courtesy.
Because the form of expression is copyrightable, avoid using exact phrases or sentences in the writing prompt.
Most writers would take a writing prompt as only the starting point for developing a story. Adding concepts, seeing what the idea looks like if parts or the whole was inverted or twisted and changed in some way, for example, choosing a different gender for the characters or setting it in different parts of the world or the universe or at different historical eras in the past or future.
If as a writers you make changes like this the idea in the writing prompt has been transformed into something that is all your own. You may still want to acknowledge the source of your inspiration, that remains your choice, but if you have made enough changes the idea could then be almost entirely yours and you don't need the writing prompt's author's permission to publish it.