Does anyone know of a website like this that has the text (not PDF) of seforim posted on their website for free?
Sefaria has a lot of text sefarim, coupled with crowdsourced English translations.
Chabad.org has Tanach here (as well as many other works of chassidus and other topics, under the tab
Learning & Values >> Texts & Writings)
http://www.sourcesonthenet.com/ (h/t msh210 in comments)
http://breslev.eip.co.il/ (Breslev seforim h/t Bachrach44)
http://chabadlibrary.org/books/ (Breslev seforim h/t Bachrach44)
http://mobile.tora.ws/ (the back end to the OnYourWay app h/t Bachrach44)